Пример 3.
Пусть на множестве A задано рефлексивное и симметричное бинарное отношение
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC4SURBVDjLY2hsbGSgFDMMHUPq62MljRkY7jIwMPwHY+OohSQZkuMmW8zAYHw3tr5esqMjjcdDhmEPyCDj6AYfogxpiDb2gRmAKsbwX9Ytp5igIXBb0ZwOM4Qol+BSjOw9goZEGTMsZJDx2JPW0cGD7jp0r2A1BB4baF4BuwLNYJyGwLyCrAGXN3AaAvNKtIdsPSxtYPMCTkNgXiGkCa8hEK/IvvHIqzck2xBssUKSIbjSxjArCojBAJ6r1eCm4k8AAAAAAElFTkSuQmCC)
(называемое отношением сходства).
Подмножество X множества A будем считать зависимым, если оно содержит два различных элемента, находящихся в отношении
.Оболочкой множества
![](data:image/png;base64,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)
служит множество
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADAAAAAZCAYAAAB3oa15AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAHHSURBVFjD7VgxbsIwFP3d6QGo4kNQd20nhJyKCzQRK0OFsjBk6GDYMtBDMHboDVh7iEpwAjgDaT6xTeLapA1RiqUifQlbif3f93vPdmA+n4PLcfEJJsm4wzxYgcdW4zi+dg6AjMmATAHoZsR510kAGAGFJdBg6SyAWUiHAGTHIt5zEkBOI4cBHCjkPAB0oyTpVALIflcY8n+lH4vni8/q7Z+OdbaIecR6BGCXTbYnvr+gABsymExPcxP2IlIazoa54I7twrhbvYq/EnEVAF0oJt7pm4wEx/moi2CLfi2XPYr694SwtzqJf8svA6HYUWVTtg1EJqevjF4AsRJp3arb7ZSuMacjbUT19IRMwrGBNQGW40oaNWKlBRrpqEqVtoGyAdMpVaKVxt06cdAQDV6NGjAlJMSZ6tWzATMVIaSU+z5Z2GjYmAvlYs0mieMbFAi2Pc/7wL4wYo/siT+UE814LcSEUeSltEvsQ/C5+5DP/vPLLQKqC+TkPqAsTtghTqzsUXtJDsTu4F3ZZ6Ey6j3Rp47EYtw/3YlP6cKJs5CJ5+2eRstzNyKkVlfgnPuA8P6taVdu6Ua2du5GpgzAsupOf5H4B3AJ8QV9AMgaUX5DpwAAAABJRU5ErkJggg==)
В этом случае можно усилить аксиому
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAAZCAYAAADaILXQAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAETSURBVEjH7VUxDoJAEFx77SXh/APZP5wU1BYYOkNFrqGwPOho/IAd/+Ap8gPegHJABPXOHHB2kkzDkWF3ZnYP0jSFXwH+5JPIm2f1DbPJI0rihqCW4N4By4Bzaxa5j+TiMu58voccgFSyMy1yzgOLUnZQdHPHY+IZNTQ5oieICY1io2nhzHUIQAXo50ajKCRCgFLXQG3yLAvXrg3FFAO1ybtkTDNwSs61DPw2VMpkyAwUZ+NOhmGTSydPhu0W4fm8GY97RzQYK75Fyk7PgiTFvCfjphj7DiOChLFtmGVr8WPhj8ybRVuvl7BWRdXIam0lE1I2nRgnFzOx3+H1vfpll0FvdhsE9C/GZHnZ+RJJ/he0Eg+bRDHHVTOYpwAAAABJRU5ErkJggg==)
отношения зависимости следующим образом:
Z
Z. Тогда оболочкой множества
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADcAAAAVCAYAAADiv3Z7AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAGlSURBVFjDY2hsbGQYrnjYemxkew4IGJExLrkh57mGaGMfoMP/AfF/CDa+E1tfLwmSizJmWIgQZ/hnHN3gM5CeALkVmxvwaqqvj5U0ZmC4y2ActRCbuKxbTvFAx059noehLIPsTY+8ekOS81yOm2wxMNbuwmKtoyONx13OeBaMP5AY5BYPGYY9DAyyb8jyHCRkGN6Aoh1smKzsSmwGkZJ3sWFyPAfKHiYmJjORA580z8GSponHWnI8hpR38WJSkzjYXGB2AacsGY89aR0dPGRVBZACBHvUE9SHw2KK85msx0qQufjsoDhd43WAcVQvTfIZMAW5ZFUblZen8YLdhlbgEeU5eB6L9kgG5TtSkg4oudCiioBWQ8hJ+j/JnkMuPOCxR0ISo4XnYPmM2CoJqyHg6EYrPNCrBKIcQsX8hpzPsJXkBD0HKpIhpRtm/qrOcjGCJU1ii254Zi8v5yW3KkC4CTVgEeIM/wl6DhoKr+FpGSnUIbGGnM5lXxNbuFBSFeByEzQ53kExA0sqGe3ykG04ES0TWvYsaNygRUpSVG6hjHZWh7PnAOmfXKReDNjbAAAAAElFTkSuQmCC)
будет множество всех элементов, находящихся в отношении сходства хотя бы с одним элементом из множества
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABUAAAATCAYAAAB/TkaLAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADUSURBVDjLY2hsbGSgNmYYNRQrrq+PlTRmYLjLwMDwH46NoxaC5Do60ng8ZBj2IMsZRzf4EO3SKGOGheiawJbmeRjKMjC8kXXLKSbZ+zDNMFfCfGHEYLQ7tr5ekqwwRXjV+C7IEIglRqdwGUh0ROW4yRaDg8DNrVZW1mNlWkcHD8WxDw8CGY89hAwk2tCGaGMfcCxTy1Cw14GRBEkFsm888uoNKTIUZCAsycBci560SDIU5DLkNAjPDEhJi2hD8bmI2CDAjGF41oOkS5xZFkl+tOhDwQAelEPh9cYI4gAAAABJRU5ErkJggg==)
.
Введенное отношение зависимости будет транзитивным тогда и только тогда, когда соответствующее бинарное отношение
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADOSURBVDjLY2hsbGSgFDPQzRAgYIRhsgypr4+VNGZguAs04B8Q/2cwjlpIkiE5brLFDAzGd2Pr6yU7OtJ4PGQY9oAMMo5u8CHKkIZoYx+YAahiDP9l3XKKCRoCtxXN6TBDiHIJLsXI3iNoSJQxw0IGGY89aR0dPOiuQ/cKVkPgsYHmFbAr0AzGaQjMK2AN5eW8oLQB9cYddG/gNATmlWgP2Xpo2viHzQs4DYF5hZAmvIZAvCL7xiOv3pBsQ7DFCkmGQAP0H3raGFxFAV0MAQDUjs+ydZnwjgAAAABJRU5ErkJggg==)
будет транзитивно, то есть является отношением эквивалентности на
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADASURBVDjLY2hsbGSgFDOMcEMaoo19jKMbfMg2pD7Pw1CWQfamR169IVmG1NfHShozMNxlYDC+G1tfL0mWIVHGsr1ubsa1ZBuS4yZbLOuWUwyiGWQ89qR1dPCQZAg4HIyjeiGuYVhIsiGgcDBiMNodW14uVV6exushw7CHwThqIUlRDLaZgeEfEv4P8hbRhsDCAT12iDYEORxQ0wjDG/SEhmEIEDBiiwGQOCilgryD1xCoIoT/oQZBvXEHm9wwLQoAmW/TaNQFHi8AAAAASUVORK5CYII=)
.
В случае, когда
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAARCAYAAADUryzEAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADLSURBVDjLY2hsbGSgBBOniIGBEYRJNqC8PI3XQ4ZhD1DzPyD+z2ActZBoA+rzPAxlGWRveuTVG4L4UcYMCxkYZN/A+HgNqK+PlTRikD2FrLgh2tgH5Arj6AYfggbkuMkWM8h47Enr6OBBNcD4bmx9vSReAzo60nhA/pZ1yylGFgd7gZgwQPc7QszoFLrtWA0AOx/qV1DUgTTLMcjdQA88rAbAne/p2WPMwHAHHH1oYYHXAGzOJyklYgt9og0AxT3I2ejxTJW8MMwNAADPJIhbQhQNBwAAAABJRU5ErkJggg==)
- отношение эквивалентности
![](data:image/png;base64,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)
будет
независимым тогда и только тогда, когда
![](data:image/png;base64,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)
множество
![](data:image/png;base64,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)
содержит не более одного элемента. Любое максимальное независимое подмножество будет содержать ровно по одному элементу из каждого класса эквивалентности
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAARCAYAAADUryzEAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADLSURBVDjLY2hsbGSgBBOniIGBEYRJNqC8PI3XQ4ZhD1DzPyD+z2ActZBoA+rzPAxlGWRveuTVG4L4UcYMCxkYZN/A+HgNqK+PlTRikD2FrLgh2tgH5Arj6AYfggbkuMkWM8h47Enr6OBBNcD4bmx9vSReAzo60nhA/pZ1yylGFgd7gZgwQPc7QszoFLrtWA0AOx/qV1DUgTTLMcjdQA88rAbAne/p2WPMwHAHHH1oYYHXAGzOJyklYgt9og0AxT3I2ejxTJW8MMwNAADPJIhbQhQNBwAAAABJRU5ErkJggg==)
.
Пример 4.
Рассмотрим четырехэлементное множество
![](data:image/png;base64,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)
.
Назовем подмножество
множества
зависимым тогда и только тогда, когда
или
. Z
. Рассмотрим подмножество
![](data:image/png;base64,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)
множества
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADASURBVDjLY2hsbGSgFDOMcEMaoo19jKMbfMg2pD7Pw1CWQfamR169IVmG1NfHShozMNxlYDC+G1tfL0mWIVHGsr1ubsa1ZBuS4yZbLOuWUwyiGWQ89qR1dPCQZAg4HIyjeiGuYVhIsiGgcDBiMNodW14uVV6exushw7CHwThqIUlRDLaZgeEfEv4P8hbRhsDCAT12iDYEORxQ0wjDG/SEhmEIEDBiiwGQOCilgryD1xCoIoT/oQZBvXEHm9wwLQoAmW/TaNQFHi8AAAAASUVORK5CYII=)
, по введенному определению оно будет независимо. Рассмотрим оболочку множества
![](data:image/png;base64,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)
и найдем оболочку оболочки нашего множества
![](data:image/png;base64,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)
. Таким образом, мы получили
![](data:image/png;base64,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)
, то есть рассмотренное нами отношение зависимости не является транзитивным.
Пример 5.
Рассмотрим произвольное множество
и ![](data:image/png;base64,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)
. Множество
будем считать зависимым, если
B (А)\ B (В), то есть
, но
. Таким образом, получили следующее транзитивное пространство зависимости:
B (А)\ B (В
. Оболочкой
![](data:image/png;base64,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)
будет множество
![](data:image/png;base64,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)
.
В частности можно рассмотреть 2 случая:
1.
![](data:image/png;base64,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)
, то есть все множества независимы, тогда
![](data:image/png;base64,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)
.
2.
B (А)
, то есть все множества, кроме пустого, будут зависимыми, в этом случае
![](data:image/png;base64,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)
.
Пример 6.
Рассмотрим произвольное множество
и его непустое конечное подмножество ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADmSURBVDjLY2hsbGSgFDMMfkOAgBEXJsqQHDfZYqDif0D8H4r/IbDsG5esaiOiXFJfHytpzMBwl8E4aiFMrKMjjcdDhmEPyGDj6AYfwobkeRjKMjC8QVYMwg3Rxj5g1yEZjtMQiJeM78bW10ti8Sphl8CdDbQNOUCjjBlA/H+ybjnFBMMEHh4oAYpqO0FDIP6WfeORV2+IERYyHnvSOjp4CBoCdjYWxbDwIOgdWHigKyTJEFxRW16exostjWAYgogBYNSWl0vBYqU6y8UIZDA4gJHSB4Yh0ID7hxsb30FPM0OoKCAFAwCz1bj1Zi/ZVAAAAABJRU5ErkJggg==)
. Введем на множестве
А следующее отношение зависимости
Z
B (А)
. Таким образом, зависимыми будут все надмножества множества
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADkSURBVDjLY2hsbGSgFDMMfkOAgBEXJsqQHDfZYqDif0D8H4r/IbDsG4+8ekOiXFJfHytpzMBwl8E4aiFMrKMjjcdDhmEPyGDj6AYfwobkeRjKMjC8QVYMwg3Rxj5g1yEZjtMQiJeM78bW10ti8Sphl8CdDbQNOUCjjBlA/H+ybjnFBMMEHh4oAcrwD91reA2B+Bs1FuBhIeOxJ62jg4egIWBnY1EMCw+C3oGFB7pCkgzBFbXl5Wm82NIIhiGIGABGbXm5FCxWqrNcjEAGgwMYKX1gGAINuH+4sfEd9DQzhIoCUjAA23i4/6VcdvAAAAAASUVORK5CYII=)
.
Если
![](data:image/png;base64,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)
, то
![](data:image/png;base64,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)
.
Если
![](data:image/png;base64,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)
, то
![](data:image/png;base64,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)
.
Если
![](data:image/png;base64,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)
, то
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGcAAAAcCAYAAAB8pKH7AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAJ4SURBVGje7VoxbsIwFHX39gBUpHcgvgKiGegBCmJlQIiFIQOqAlvO0I2xZygn6A0qwQnKGZLGSeyYKIl/YseB1kh/SmK/7/ff87cF2u12yMR1hlmEWyUn+t0pm0jhWIacZEFDheSEZsEVkUMqPYpAITmBUY86cpRXulGPBnJSVbGoet4VOU0x6sZVibGupXnerIcROpKFZoEne/LM9+f3Th8duGcBnm7Huq1NBmObsRxZ63QN2NxZWOfhYmNDyBFW+ASjPXkvn5i3cgYWQj/WaLnu2tqaYtRQOCdaLPmCuSjmpo1AmuCZn4RMbCP7c+Z5vWtoDJpi1IEpXzDbKR7zCi8jB1TZGdv4RBJNJrW/IEnrUo8MRrJYdZRFLAtij4m1JXgKLE+onLAOoHjA0ejNspyPue/fy3ZtVRtmE8U1xdgGOaxYInXwOaX2G+TnkzrbMNvoOwdo0lXWRkFWBif7NjG2QQ7bby5zCvGr9yJspevaDfPJmonrtLamGOPvahQCKSwROQkW6+ysvAEEn5ylReCTar+cUIYclbYmgzFWHNAGiV09P+F30V4W4ygggVqvyNZAnRQZjA5Ema9zViizT5W2JosRqgaoyuh+U2SVIHIgmzUBzA9S1Ld3aWmqMGb7lfVdpThoB1jWQrvu/KHojAMmhyRcVn1Zt5GccCHKa+neTinG/IE132FlY+KjiBju3dPMdR/p95vF0I6blRJHEHZSDByzlgwMd00SFD2vezUkebBThlFw9cIC0s2lBVNh1eVYurAdczMNXSudVyxtqObfkdPivmBUY8j5++QEqv/gYSxNETmUIIXkGGJUkmPCkGPCkHN78Quq4CSPl8yJ1wAAAABJRU5ErkJggg==)
.
Получаем транзитивное пространство зависимости.
Пример 7.
Подпространство пространства зависимости
Z
. Рассмотрим ![](data:image/png;base64,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)
, где действует то же отношение зависимости
Z. Тогда получим индуцированное пространство зависимости
Z
B
. В этом случае зависимыми будут только те подмножества множества
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADmSURBVDjLY2hsbGSgFDMMfkOAgBEXJsqQHDfZYqDif0D8H4r/IbDsG5esaiOiXFJfHytpzMBwl8E4aiFMrKMjjcdDhmEPyGDj6AYfwobkeRjKMjC8QVYMwg3Rxj5g1yEZjtMQiJeM78bW10ti8Sphl8CdDbQNOUCjjBlA/H+ybjnFBMMEHh4oAYpqO0FDIP6WfeORV2+IERYyHnvSOjp4CBoCdjYWxbDwIOgdWHigKyTJEFxRW16exostjWAYgogBYNSWl0vBYqU6y8UIZDA4gJHSB4Yh0ID7hxsb30FPM0OoKCAFAwCz1bj1Zi/ZVAAAAABJRU5ErkJggg==)
, которые были зависимы в пространстве
Z
. И если пространство
Z ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAAcCAYAAACDBoQZAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADrSURBVEjHY2hsbGSgFmYYMMPS0tIMqGJYeXm5AA8Pz/OOjg4OqrhMR0dneWhoaARVDIuNjfVQUVHZThXDQF4UEBC4D/SyBFVi09zcvN/Nza2AKoYBY9RCRkbmONXSmYiIyPW8vDwNqhjm4uJSYWNj004Vw4qLixVAEUG17KSgoLA/ISHBgSqGAb1aBDRwL1UMMzAwmO/j45NKsWGgxAvKp6D8SrFhoPwJyqdUiQBQ/gTlU4oNA+VLkBepkgNA+RKUP6limLi4+HlcJS7JxTbIMKqUZ6D8CMqXVDEMmrYkqFKh4CoUB77epKthANFVGTHVFSNEAAAAAElFTkSuQmCC)
транзитивно, то транзитивным будет и подпространство
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAATCAYAAAB2pebxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADmSURBVDjLY2hsbGSgFDMMfkOAgBEXJsqQHDfZYqDif0D8H4r/IbDsG5esaiOiXFJfHytpzMBwl8E4aiFMrKMjjcdDhmEPyGDj6AYfwobkeRjKMjC8QVYMwg3Rxj5g1yEZjtMQiJeM78bW10ti8Sphl8CdDbQNOUCjjBlA/H+ybjnFBMMEHh4oAYpqO0FDIP6WfeORV2+IERYyHnvSOjp4CBoCdjYWxbDwIOgdWHigKyTJEFxRW16exostjWAYgogBYNSWl0vBYqU6y8UIZDA4gJHSB4Yh0ID7hxsb30FPM0OoKCAFAwCz1bj1Zi/ZVAAAAABJRU5ErkJggg==)
.