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 Th51:
  crossover(p1,p2,n1,n2,n3,n4,n5)=crossover(p1,p2,n2,n1,n3,n4,n5)
& crossover(p1,p2,n1,n2,n3,n4,n5)=crossover(p1,p2,n3,n2,n1,n4,n5) & crossover(
p1,p2,n1,n2,n3,n4,n5)=crossover(p1,p2,n4,n2,n3,n1,n5) & crossover(p1,p2,n1,n2,
  n3,n4,n5)=crossover(p1,p2,n5,n2,n3,n4,n1)
proof
A1: crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(crossover(p1,p2,n3,n2,n1,n4),
  crossover(p2,p1,n1,n2,n3,n4),n5) by Th37
    .=crossover(crossover(p1,p2,n3,n2,n1,n4), crossover(p2,p1,n3,n2,n1,n4),
  n5) by Th37;
A2: crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(crossover(p1,p2,n4,n2,n3,n1),
  crossover(p2,p1,n1,n2,n3,n4),n5) by Th37
    .=crossover(crossover(p1,p2,n4,n2,n3,n1), crossover(p2,p1,n4,n2,n3,n1),
  n5) by Th37;
A3: crossover(p1,p2,n1,n2,n3,n4,n5)=crossover(p1,p2,n5,n2,n3,n4,n1)
  proof
    set q2=crossover(p2,p1,n2,n3,n4);
    set q1=crossover(p1,p2,n2,n3,n4);
A4: crossover(p1,p2,n2,n3,n4,n5)=crossover(q1,q2,n5);
    crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(crossover(p1,p2,n2,n3,n4,n1
    ), crossover(p2,p1,n1,n2,n3,n4),n5) by Th37
      .=crossover(crossover(p1,p2,n2,n3,n4,n1), crossover(p2,p1,n2,n3,n4,n1)
    ,n5) by Th37
      .=crossover(q1,q2,n1,n5)
      .=crossover(q1,q2,n5,n1) by Th13
      .=crossover(crossover(p1,p2,n5,n2,n3,n4), crossover(p2,p1,n2,n3,n4,n5)
    ,n1) by A4,Th37
      .=crossover(crossover(p1,p2,n5,n2,n3,n4), crossover(p2,p1,n5,n2,n3,n4)
    ,n1) by Th37;
    hence thesis;
  end;
  crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(crossover(p1,p2,n2,n1,n3,n4),
  crossover(p2,p1,n1,n2,n3,n4),n5) by Th37
    .=crossover(crossover(p1,p2,n2,n1,n3,n4), crossover(p2,p1,n2,n1,n3,n4),
  n5) by Th37;
  hence thesis by A1,A2,A3;
end;
