let n1, n2, n3 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,n3) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2) )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,n3) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2) )
let p1, p2 be Individual of S; :: thesis: ( crossover (p1,p2,0,n2,n3) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,n3) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2) )
crossover (p1,p2,0,n2,n3) = crossover ((crossover (p2,p1,n2)),(crossover (p2,p1,0,n2)),n3) by Th7
.= crossover ((crossover (p2,p1,n2)),(crossover (p1,p2,n2)),n3) by Th7 ;
hence crossover (p1,p2,0,n2,n3) = crossover (p2,p1,n2,n3) ; :: thesis: ( crossover (p1,p2,n1,0,n3) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2) )
crossover (p1,p2,n1,0,n3) = crossover ((crossover (p2,p1,n1)),(crossover (p2,p1,n1,0)),n3) by Th8
.= crossover ((crossover (p2,p1,n1)),(crossover (p1,p2,n1)),n3) by Th8 ;
hence crossover (p1,p2,n1,0,n3) = crossover (p2,p1,n1,n3) ; :: thesis: crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2)
crossover (p1,p2,n1,n2,0) = crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),0) ;
hence crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2) by Th4; :: thesis: verum