let n be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds crossover p1,p2,n,0 = crossover p2,p1,n
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds crossover p1,p2,n,0 = crossover p2,p1,n
let p1, p2 be Individual of S; :: thesis: crossover p1,p2,n,0 = crossover p2,p1,n
reconsider q1 = crossover p1,p2,n as Individual of S ;
reconsider q2 = crossover p2,p1,n as Individual of S ;
crossover p1,p2,n,0 = crossover q1,q2,0
.= q2 by Th4 ;
hence crossover p1,p2,n,0 = crossover p2,p1,n ; :: thesis: verum