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