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 Th52:
  crossover(p1,p2,n1,n1,n3,n4,n5)=crossover(p1,p2,n3,n4,n5) &
crossover(p1,p2,n1,n2,n1,n4,n5)=crossover(p1,p2,n2,n4,n5) & crossover(p1,p2,n1,
  n2,n3,n1,n5)=crossover(p1,p2,n2,n3,n5) & crossover(p1,p2,n1,n2,n3,n4,n1)=
  crossover(p1,p2,n2,n3,n4)
proof
  set q1=crossover(p1,p2,n2,n3,n4);
  set q2=crossover(p2,p1,n2,n3,n4);
  crossover(p1,p2,n1,n1,n3,n4,n5) =crossover(crossover(p1,p2,n3,n4),
  crossover(p2,p1,n1,n1,n3,n4),n5) by Th38
    .=crossover(crossover(p1,p2,n3,n4),crossover(p2,p1,n3,n4),n5) by Th38;
  hence crossover(p1,p2,n1,n1,n3,n4,n5)=crossover(p1,p2,n3,n4,n5);
  crossover(p1,p2,n1,n2,n1,n4,n5) =crossover(crossover(p1,p2,n2,n4),
  crossover(p2,p1,n1,n2,n1,n4),n5) by Th38
    .=crossover(crossover(p1,p2,n2,n4),crossover(p2,p1,n2,n4),n5) by Th38;
  hence crossover(p1,p2,n1,n2,n1,n4,n5)=crossover(p1,p2,n2,n4,n5);
  crossover(p1,p2,n1,n2,n3,n1,n5) =crossover(crossover(p1,p2,n2,n3),
  crossover(p2,p1,n1,n2,n3,n1),n5) by Th38
    .=crossover(crossover(p1,p2,n2,n3),crossover(p2,p1,n2,n3),n5) by Th38;
  hence crossover(p1,p2,n1,n2,n3,n1,n5)=crossover(p1,p2,n2,n3,n5);
  crossover(p1,p2,n1,n2,n3,n4,n1) =crossover(crossover(p1,p2,n2,n3,n4,n1),
  crossover(p2,p1,n1,n2,n3,n4),n1) by Th37
    .=crossover(crossover(p1,p2,n2,n3,n4,n1), crossover(p2,p1,n2,n3,n4,n1),
  n1) by Th37
    .=crossover(q1,q2,n1,n1);
  hence thesis by Th12;
end;
