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,n1,n1,n3) = crossover (p1,p2,n3) & crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n2) & crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n1) )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3) = crossover (p1,p2,n3) & crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n2) & crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n1) )
let p1, p2 be Individual of S; :: thesis: ( crossover (p1,p2,n1,n1,n3) = crossover (p1,p2,n3) & crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n2) & crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n1) )
crossover (p1,p2,n1,n1,n3) = crossover (p1,(crossover (p2,p1,n1,n1)),n3) by Th12;
hence crossover (p1,p2,n1,n1,n3) = crossover (p1,p2,n3) by Th12; :: thesis: ( crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n2) & crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n1) )
crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n1,n1,n2) by Th25
.= crossover (p1,(crossover (p2,p1,n1,n1)),n2) by Th12 ;
hence crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n2) by Th12; :: thesis: crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n1)
thus crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n2,n2,n1) by Th26
.= crossover (p1,(crossover (p2,p1,n2,n2)),n1) by Th12
.= crossover (p1,p2,n1) by Th12 ; :: thesis: verum