let n1, n2, n3 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,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; :: thesis: 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; :: thesis: ( 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; :: thesis: ( 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; :: thesis: 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; :: thesis: verum