let n1, n2, n3 be Element of NAT ; for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,n3) = crossover (p1,p2,n3) & crossover (p1,p2,n1,0,0) = crossover (p1,p2,n1) & crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2) )
let S be Gene-Set; for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,n3) = crossover (p1,p2,n3) & crossover (p1,p2,n1,0,0) = crossover (p1,p2,n1) & crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2) )
let p1, p2 be Individual of S; ( crossover (p1,p2,0,0,n3) = crossover (p1,p2,n3) & crossover (p1,p2,n1,0,0) = crossover (p1,p2,n1) & crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2) )
crossover (p1,p2,0,0,n3) = crossover (p1,(crossover (p2,p1,0,0)),n3)
by Th12;
hence
crossover (p1,p2,0,0,n3) = crossover (p1,p2,n3)
by Th12; ( crossover (p1,p2,n1,0,0) = crossover (p1,p2,n1) & crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2) )
crossover (p1,p2,n1,0,0) = crossover ((crossover (p2,p1,n1)),(crossover (p1,p2,n1)),0)
by Th8;
hence
crossover (p1,p2,n1,0,0) = crossover (p1,p2,n1)
by Th4; crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2)
crossover (p1,p2,0,n2,0) = crossover ((crossover (p2,p1,n2)),(crossover (p1,p2,n2)),0)
by Th7;
hence
crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2)
by Th4; verum