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
  crossover(p1,p2,0,0,n3) = crossover(p1,p2,n3) & crossover(p1,p2,n1,0,0
  ) = crossover(p1,p2,n1) & crossover(p1,p2,0,n2,0) = crossover(p1,p2,n2)
proof
  crossover(p1,p2,0,0,n3) = crossover(p1,crossover(p2,p1,0,0),n3) by Th12;
  hence crossover(p1,p2,0,0,n3) = crossover(p1,p2,n3) by Th12;
  crossover(p1,p2,n1,0,0) = crossover(crossover(p2,p1,n1),crossover(p1,p2,
  n1),0) by Th8;
  hence crossover(p1,p2,n1,0,0) = crossover(p1,p2,n1) by Th4;
  crossover(p1,p2,0,n2,0) = crossover(crossover(p2,p1,n2),crossover(p1,p2,
  n2),0) by Th7;
  hence thesis by Th4;
end;
