let n3, n1, n2 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