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 Th31:
  crossover(p1,p2,n1,0,0,0) = crossover(p2,p1,n1) & crossover(p1,
p2,0,n2,0,0) = crossover(p2,p1,n2) & crossover(p1,p2,0,0,n3,0) = crossover(p2,
  p1,n3) & crossover(p1,p2,0,0,0,n4) = crossover(p2,p1,n4)
proof
  crossover(p1,p2,n1,0,0,0) = crossover(p1,p2,n1,0) by Th30;
  hence crossover(p1,p2,n1,0,0,0) = crossover(p2,p1,n1) by Th8;
  crossover(p1,p2,0,n2,0,0) = crossover(p1,p2,0,n2) by Th30;
  hence crossover(p1,p2,0,n2,0,0) = crossover(p2,p1,n2) by Th7;
  crossover(p1,p2,0,0,n3,0) = crossover(p1,p2,0,n3) by Th30;
  hence crossover(p1,p2,0,0,n3,0) = crossover(p2,p1,n3) by Th7;
  crossover(p1,p2,0,0,0,n4) = crossover(p1,p2,0,n4) by Th30;
  hence thesis by Th7;
end;
