let n1, n2, n3, n4 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) = crossover (p1,p2,n3,n4) & crossover (p1,p2,n1,n2,n1,n4) = crossover (p1,p2,n2,n4) & crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n4) = crossover (p1,p2,n3,n4) & crossover (p1,p2,n1,n2,n1,n4) = crossover (p1,p2,n2,n4) & crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
let p1, p2 be Individual of S; :: thesis: ( crossover (p1,p2,n1,n1,n3,n4) = crossover (p1,p2,n3,n4) & crossover (p1,p2,n1,n2,n1,n4) = crossover (p1,p2,n2,n4) & crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
crossover (p1,p2,n1,n1,n3,n4) = crossover ((crossover (p1,p2,n3)),(crossover (p2,p1,n1,n1,n3)),n4) by Th27
.= crossover ((crossover (p1,p2,n3)),(crossover (p2,p1,n3)),n4) by Th27 ;
hence crossover (p1,p2,n1,n1,n3,n4) = crossover (p1,p2,n3,n4) ; :: thesis: ( crossover (p1,p2,n1,n2,n1,n4) = crossover (p1,p2,n2,n4) & crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
crossover (p1,p2,n1,n2,n1,n4) = crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n1,n2,n1)),n4) by Th27
.= crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n2)),n4) by Th27 ;
hence crossover (p1,p2,n1,n2,n1,n4) = crossover (p1,p2,n2,n4) ; :: thesis: ( crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n1,n1,n2,n3) by Th37
.= crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n1,n1,n2)),n3) by Th27
.= crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n2)),n3) by Th27 ;
hence crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n2,n3) ; :: thesis: ( crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
crossover (p1,p2,n1,n2,n2,n4) = crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1,n2,n2)),n4) by Th27
.= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n4) by Th27 ;
hence crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) ; :: thesis: ( crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n2,n2,n3) by Th37
.= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1,n2,n2)),n3) by Th27
.= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n3) by Th27 ;
hence crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) ; :: thesis: crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2)
crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n3,n3,n2) by Th37
.= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1,n3,n3)),n2) by Th27
.= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n2) by Th27 ;
hence crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) ; :: thesis: verum