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,n4,n5,n6)=crossover(p1,p2,n3,n4,n5,n6) &
crossover(p1,p2,n1,n2,n1,n4,n5,n6)=crossover(p1,p2,n2,n4,n5,n6) & crossover(p1,
p2,n1,n2,n3,n1,n5,n6)=crossover(p1,p2,n2,n3,n5,n6) & crossover(p1,p2,n1,n2,n3,
  n4,n1,n6)=crossover(p1,p2,n2,n3,n4,n6) & crossover(p1,p2,n1,n2,n3,n4,n5,n1)=
  crossover(p1,p2,n2,n3,n4,n5)
proof
  crossover(p1,p2,n1,n1,n3,n4,n5,n6) =crossover(crossover(p1,p2,n3,n4,n5),
  crossover(p2,p1,n1,n1,n3,n4,n5),n6) by Th52
    .=crossover(crossover(p1,p2,n3,n4,n5), crossover(p2,p1,n3,n4,n5),n6) by
Th52;
  hence crossover(p1,p2,n1,n1,n3,n4,n5,n6)=crossover(p1,p2,n3,n4,n5,n6);
  crossover(p1,p2,n1,n2,n1,n4,n5,n6) =crossover(crossover(p1,p2,n2,n4,n5),
  crossover(p2,p1,n1,n2,n1,n4,n5),n6) by Th52
    .=crossover(crossover(p1,p2,n2,n4,n5), crossover(p2,p1,n2,n4,n5),n6) by
Th52;
  hence crossover(p1,p2,n1,n2,n1,n4,n5,n6)=crossover(p1,p2,n2,n4,n5,n6);
  crossover(p1,p2,n1,n2,n3,n1,n5,n6) =crossover(crossover(p1,p2,n2,n3,n5),
  crossover(p2,p1,n1,n2,n3,n1,n5),n6) by Th52
    .=crossover(crossover(p1,p2,n2,n3,n5), crossover(p2,p1,n2,n3,n5),n6) by
Th52;
  hence crossover(p1,p2,n1,n2,n3,n1,n5,n6)=crossover(p1,p2,n2,n3,n5,n6);
  crossover(p1,p2,n1,n2,n3,n4,n1,n6) =crossover(crossover(p1,p2,n2,n3,n4),
  crossover(p2,p1,n1,n2,n3,n4,n1),n6) by Th52
    .=crossover(crossover(p1,p2,n2,n3,n4), crossover(p2,p1,n2,n3,n4),n6) by
Th52;
  hence crossover(p1,p2,n1,n2,n3,n4,n1,n6)=crossover(p1,p2,n2,n3,n4,n6);
  crossover(p1,p2,n1,n2,n3,n4,n5,n1) =crossover(p1,p2,n5,n2,n3,n4,n1,n1)
  by Th56
    .=crossover(p1,p2,n1,n2,n3,n4,n1,n5) by Th56
    .=crossover(crossover(p1,p2,n2,n3,n4), crossover(p2,p1,n1,n2,n3,n4,n1),
  n5) by Th52
    .=crossover(crossover(p1,p2,n2,n3,n4), crossover(p2,p1,n2,n3,n4),n5) by
Th52;
  hence thesis;
end;
