let n1, n2, n3, n4, n5, n6 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n4,n5,n6) = crossover (p1,p2,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover (p1,p2,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover (p1,p2,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n4,n5,n6) = crossover (p1,p2,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover (p1,p2,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover (p1,p2,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) )
let p1, p2 be Individual of S; :: thesis: ( crossover (p1,p2,n1,n1,n3,n4,n5,n6) = crossover (p1,p2,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover (p1,p2,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover (p1,p2,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) )
crossover (p1,p2,n1,n1,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n3,n4,n5)),(crossover (p2,p1,n1,n1,n3,n4,n5)),n6) by Th52
.= crossover ((crossover (p1,p2,n3,n4,n5)),(crossover (p2,p1,n3,n4,n5)),n6) by Th52 ;
hence crossover (p1,p2,n1,n1,n3,n4,n5,n6) = crossover (p1,p2,n3,n4,n5,n6) ; :: thesis: ( crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover (p1,p2,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover (p1,p2,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) )
crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover ((crossover (p1,p2,n2,n4,n5)),(crossover (p2,p1,n1,n2,n1,n4,n5)),n6) by Th52
.= crossover ((crossover (p1,p2,n2,n4,n5)),(crossover (p2,p1,n2,n4,n5)),n6) by Th52 ;
hence crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover (p1,p2,n2,n4,n5,n6) ; :: thesis: ( crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover (p1,p2,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) )
crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover ((crossover (p1,p2,n2,n3,n5)),(crossover (p2,p1,n1,n2,n3,n1,n5)),n6) by Th52
.= crossover ((crossover (p1,p2,n2,n3,n5)),(crossover (p2,p1,n2,n3,n5)),n6) by Th52 ;
hence crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover (p1,p2,n2,n3,n5,n6) ; :: thesis: ( crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) )
crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4,n1)),n6) by Th52
.= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n6) by Th52 ;
hence crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) ; :: thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5)
crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n5,n2,n3,n4,n1,n1) by Th56
.= crossover (p1,p2,n1,n2,n3,n4,n1,n5) by Th56
.= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4,n1)),n5) by Th52
.= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n5) by Th52 ;
hence crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) ; :: thesis: verum