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,n1,n1,n3,n3) = p1 & crossover(p1,p2,n1,n2,n1,n2) = p1
  & crossover(p1,p2,n1,n2,n2,n1) = p1
proof
  crossover(p1,p2,n1,n1,n3,n3)=crossover(p1,p2,n3,n3) by Th38;
  hence crossover(p1,p2,n1,n1,n3,n3) = p1 by Th12;
  crossover(p1,p2,n1,n2,n1,n2)=crossover(p1,p2,n2,n2) by Th38;
  hence crossover(p1,p2,n1,n2,n1,n2) = p1 by Th12;
  crossover(p1,p2,n1,n2,n2,n1)=crossover(p1,p2,n2,n2) by Th38;
  hence thesis by Th12;
end;
