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 Th25:
  crossover(p1,p2,n1,n2,n3) = crossover(p1,p2,n2,n1,n3) &
  crossover(p1,p2,n1,n2,n3) = crossover(p1,p2,n1,n3,n2)
proof
  set q1=crossover(p1,p2,n1);
  set q2=crossover(p2,p1,n1);
  crossover(p1,p2,n1,n2,n3) = crossover(crossover(p1,p2,n2,n1),crossover(
  p2,p1,n1,n2),n3) by Th13
    .= crossover(crossover(p1,p2,n2,n1),crossover(p2,p1,n2,n1),n3) by Th13;
  hence crossover(p1,p2,n1,n2,n3) = crossover(p1,p2,n2,n1,n3);
  crossover(p1,p2,n1,n2,n3) = crossover(q1,q2,n2,n3)
    .= crossover(q1,q2,n3,n2) by Th13
    .= crossover(crossover(p1,p2,n1,n3),crossover(p2,p1,n1,n3),n2);
  hence thesis;
end;
