let n2, n3, n4, n5, n6, n1 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3,n4,n5,n6) = crossover (p2,p1,n2,n3,n4,n5,n6) & crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover (p2,p1,n1,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover (p2,p1,n1,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3,n4,n5,n6) = crossover (p2,p1,n2,n3,n4,n5,n6) & crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover (p2,p1,n1,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover (p2,p1,n1,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
let p1, p2 be Individual of S; :: thesis: ( crossover (p1,p2,0,n2,n3,n4,n5,n6) = crossover (p2,p1,n2,n3,n4,n5,n6) & crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover (p2,p1,n1,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover (p2,p1,n1,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
crossover (p1,p2,0,n2,n3,n4,n5,n6) = crossover ((crossover (p2,p1,n2,n3,n4,n5)),(crossover (p2,p1,0,n2,n3,n4,n5)),n6) by Th41
.= crossover ((crossover (p2,p1,n2,n3,n4,n5)),(crossover (p1,p2,n2,n3,n4,n5)),n6) by Th41 ;
hence crossover (p1,p2,0,n2,n3,n4,n5,n6) = crossover (p2,p1,n2,n3,n4,n5,n6) ; :: thesis: ( crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover (p2,p1,n1,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover (p2,p1,n1,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover ((crossover (p2,p1,n1,n3,n4,n5)),(crossover (p2,p1,n1,0,n3,n4,n5)),n6) by Th41
.= crossover ((crossover (p2,p1,n1,n3,n4,n5)),(crossover (p1,p2,n1,n3,n4,n5)),n6) by Th41 ;
hence crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover (p2,p1,n1,n3,n4,n5,n6) ; :: thesis: ( crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover (p2,p1,n1,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover ((crossover (p2,p1,n1,n2,n4,n5)),(crossover (p2,p1,n1,n2,0,n4,n5)),n6) by Th41
.= crossover ((crossover (p2,p1,n1,n2,n4,n5)),(crossover (p1,p2,n1,n2,n4,n5)),n6) by Th41 ;
hence crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover (p2,p1,n1,n2,n4,n5,n6) ; :: thesis: ( crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover ((crossover (p2,p1,n1,n2,n3,n5)),(crossover (p2,p1,n1,n2,n3,0,n5)),n6) by Th41
.= crossover ((crossover (p2,p1,n1,n2,n3,n5)),(crossover (p1,p2,n1,n2,n3,n5)),n6) by Th41 ;
hence crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) ; :: thesis: ( crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover ((crossover (p2,p1,n1,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4,0)),n6) by Th41
.= crossover ((crossover (p2,p1,n1,n2,n3,n4)),(crossover (p1,p2,n1,n2,n3,n4)),n6) by Th41 ;
hence crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) ; :: thesis: crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5)
thus crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover ((crossover (p1,p2,n1,n2,n3,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),0)
.= crossover (p2,p1,n1,n2,n3,n4,n5) by Th4 ; :: thesis: verum