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 Th15:
  crossover(p1,p2,0,n2,n3) = crossover(p2,p1,n2,n3) & crossover(p1
,p2,n1,0,n3) = crossover(p2,p1,n1,n3) & crossover(p1,p2,n1,n2,0) = crossover(p2
  ,p1,n1,n2)
proof
  crossover(p1,p2,0,n2,n3) = crossover(crossover(p2,p1,n2),crossover(p2,p1
  ,0,n2),n3) by Th7
    .= crossover(crossover(p2,p1,n2),crossover(p1,p2,n2),n3) by Th7;
  hence crossover(p1,p2,0,n2,n3) = crossover(p2,p1,n2,n3);
  crossover(p1,p2,n1,0,n3) = crossover(crossover(p2,p1,n1),crossover(p2,p1
  ,n1,0),n3) by Th8
    .= crossover(crossover(p2,p1,n1),crossover(p1,p2,n1),n3) by Th8;
  hence crossover(p1,p2,n1,0,n3) = crossover(p2,p1,n1,n3);
  crossover(p1,p2,n1,n2,0) = crossover(crossover(p1,p2,n1,n2),crossover(p2
  ,p1,n1,n2),0);
  hence thesis by Th4;
end;
