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 Th29:
  crossover(p1,p2,0,n2,n3,n4) = crossover(p2,p1,n2,n3,n4) &
crossover(p1,p2,n1,0,n3,n4) = crossover(p2,p1,n1,n3,n4) & crossover(p1,p2,n1,n2
,0,n4) = crossover(p2,p1,n1,n2,n4) & crossover(p1,p2,n1,n2,n3,0) = crossover(p2
  ,p1,n1,n2,n3)
proof
  crossover(p1,p2,0,n2,n3,n4) =crossover(crossover(p2,p1,n2,n3),crossover(
  p2,p1,0,n2,n3),n4) by Th15
    .=crossover(crossover(p2,p1,n2,n3),crossover(p1,p2,n2,n3),n4) by Th15;
  hence crossover(p1,p2,0,n2,n3,n4) = crossover(p2,p1,n2,n3,n4);
  crossover(p1,p2,n1,0,n3,n4) =crossover(crossover(p2,p1,n1,n3),crossover(
  p2,p1,n1,0,n3),n4) by Th15
    .=crossover(crossover(p2,p1,n1,n3),crossover(p1,p2,n1,n3),n4) by Th15;
  hence crossover(p1,p2,n1,0,n3,n4) = crossover(p2,p1,n1,n3,n4);
  crossover(p1,p2,n1,n2,0,n4) =crossover(crossover(p2,p1,n1,n2),crossover(
  p2,p1,n1,n2,0),n4) by Th15
    .=crossover(crossover(p2,p1,n1,n2),crossover(p1,p2,n1,n2),n4) by Th15;
  hence crossover(p1,p2,n1,n2,0,n4) = crossover(p2,p1,n1,n2,n4);
  crossover(p1,p2,n1,n2,n3,0) =crossover(crossover(p1,p2,n1,n2,n3),
  crossover(p2,p1,n1,n2,n3),0);
  hence thesis by Th4;
end;
