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 Th18:
  n1 >= len p1 implies crossover(p1,p2,n1,n2,n3) = crossover(p1,p2 ,n2,n3)
proof
  assume
A1: n1 >= len p1;
  then n1 >= len S by Def1;
  then
A2: n1 >= len p2 by Def1;
  crossover(p1,p2,n1,n2,n3) = crossover(crossover(p1,p2,n2),crossover(p2,
  p1,n1,n2),n3) by A1,Th9
    .= crossover(crossover(p1,p2,n2),crossover(p2,p1,n2),n3) by A2,Th9;
  hence thesis;
end;
