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 Th27:
  crossover(p1,p2,n1,n1,n3)=crossover(p1,p2,n3) & crossover(p1,p2,
  n1,n2,n1)=crossover(p1,p2,n2) & crossover(p1,p2,n1,n2,n2)=crossover(p1,p2,n1)
proof
  crossover(p1,p2,n1,n1,n3) =crossover(p1,crossover(p2,p1,n1,n1),n3) by Th12;
  hence crossover(p1,p2,n1,n1,n3)=crossover(p1,p2,n3) by Th12;
  crossover(p1,p2,n1,n2,n1) = crossover(p1,p2,n1,n1,n2) by Th25
    .=crossover(p1,crossover(p2,p1,n1,n1),n2) by Th12;
  hence crossover(p1,p2,n1,n2,n1)=crossover(p1,p2,n2) by Th12;
  thus crossover(p1,p2,n1,n2,n2) =crossover(p1,p2,n2,n2,n1) by Th26
    .=crossover(p1,crossover(p2,p1,n2,n2),n1) by Th12
    .=crossover(p1,p2,n1) by Th12;
end;
