reserve D for non empty set;
reserve f1,f2 for FinSequence of D;
reserve i,n,n1,n2,n3,n4,n5,n6 for Element of NAT;
reserve S for Gene-Set;
reserve p1,p2 for Individual of S;

theorem Th8:
  crossover(p1,p2,n,0) = crossover(p2,p1,n)
proof
  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 thesis;
end;
