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
  n1>=len p1 & n2>=len p1 & n3>=len p1 & n4>=len p1 & n5>=len p1 implies
  crossover(p1,p2,n1,n2,n3,n4,n5)=p1
proof
  assume that
A1: n1>=len p1 & n2>=len p1 & n3>=len p1 & n4>=len p1 and
A2: n5>=len p1;
  crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(p1,crossover(p2,p1,n1,n2,n3,
  n4),n5) by A1,Th36;
  hence thesis by A2,Th5;
end;
