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 Th38:
  crossover(p1,p2,n1,n1,n3,n4) = crossover(p1,p2,n3,n4) &
crossover(p1,p2,n1,n2,n1,n4) = crossover(p1,p2,n2,n4) & crossover(p1,p2,n1,n2,
n3,n1) = crossover(p1,p2,n2,n3) & crossover(p1,p2,n1,n2,n2,n4) = crossover(p1,
p2,n1,n4) & crossover(p1,p2,n1,n2,n3,n2) = crossover(p1,p2,n1,n3) & crossover(
  p1,p2,n1,n2,n3,n3) = crossover(p1,p2,n1,n2)
proof
  crossover(p1,p2,n1,n1,n3,n4) =crossover(crossover(p1,p2,n3),crossover(p2
  ,p1,n1,n1,n3),n4) by Th27
    .=crossover(crossover(p1,p2,n3),crossover(p2,p1,n3),n4) by Th27;
  hence crossover(p1,p2,n1,n1,n3,n4) = crossover(p1,p2,n3,n4);
  crossover(p1,p2,n1,n2,n1,n4) =crossover(crossover(p1,p2,n2),crossover(p2
  ,p1,n1,n2,n1),n4) by Th27
    .=crossover(crossover(p1,p2,n2),crossover(p2,p1,n2),n4) by Th27;
  hence crossover(p1,p2,n1,n2,n1,n4) = crossover(p1,p2,n2,n4);
  crossover(p1,p2,n1,n2,n3,n1) = crossover(p1,p2,n1,n1,n2,n3) by Th37
    .=crossover(crossover(p1,p2,n2),crossover(p2,p1,n1,n1,n2),n3) by Th27
    .=crossover(crossover(p1,p2,n2),crossover(p2,p1,n2),n3) by Th27;
  hence crossover(p1,p2,n1,n2,n3,n1) = crossover(p1,p2,n2,n3);
  crossover(p1,p2,n1,n2,n2,n4) =crossover(crossover(p1,p2,n1),crossover(p2
  ,p1,n1,n2,n2),n4) by Th27
    .=crossover(crossover(p1,p2,n1),crossover(p2,p1,n1),n4) by Th27;
  hence crossover(p1,p2,n1,n2,n2,n4) = crossover(p1,p2,n1,n4);
  crossover(p1,p2,n1,n2,n3,n2) = crossover(p1,p2,n1,n2,n2,n3) by Th37
    .=crossover(crossover(p1,p2,n1),crossover(p2,p1,n1,n2,n2),n3) by Th27
    .=crossover(crossover(p1,p2,n1),crossover(p2,p1,n1),n3) by Th27;
  hence crossover(p1,p2,n1,n2,n3,n2) = crossover(p1,p2,n1,n3);
  crossover(p1,p2,n1,n2,n3,n3) = crossover(p1,p2,n1,n3,n3,n2) by Th37
    .=crossover(crossover(p1,p2,n1),crossover(p2,p1,n1,n3,n3),n2) by Th27
    .=crossover(crossover(p1,p2,n1),crossover(p2,p1,n1),n2) by Th27;
  hence thesis;
end;
