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 Th56:
  crossover(p1,p2,n1,n2,n3,n4,n5,n6)=crossover(p1,p2,n2,n1,n3,n4,
n5,n6) & crossover(p1,p2,n1,n2,n3,n4,n5,n6)=crossover(p1,p2,n3,n2,n1,n4,n5,n6)
  & crossover(p1,p2,n1,n2,n3,n4,n5,n6)=crossover(p1,p2,n4,n2,n3,n1,n5,n6) &
  crossover(p1,p2,n1,n2,n3,n4,n5,n6)=crossover(p1,p2,n5,n2,n3,n4,n1,n6) &
  crossover(p1,p2,n1,n2,n3,n4,n5,n6)=crossover(p1,p2,n6,n2,n3,n4,n5,n1)
proof
  set q1=crossover(p1,p2,n5,n2,n3,n4);
  set q2=crossover(p2,p1,n5,n2,n3,n4);
A1: crossover(p1,p2,n5,n2,n3,n4,n6)=crossover(q1,q2,n6);
  crossover(p1,p2,n1,n2,n3,n4,n5,n6) =crossover(crossover(p1,p2,n2,n1,n3,
  n4,n5), crossover(p2,p1,n1,n2,n3,n4,n5),n6) by Th51
    .=crossover(crossover(p1,p2,n2,n1,n3,n4,n5), crossover(p2,p1,n2,n1,n3,n4
  ,n5),n6) by Th51;
  hence crossover(p1,p2,n1,n2,n3,n4,n5,n6)=crossover(p1,p2,n2,n1,n3,n4,n5,n6);
  crossover(p1,p2,n1,n2,n3,n4,n5,n6) =crossover(crossover(p1,p2,n3,n2,n1,
  n4,n5), crossover(p2,p1,n1,n2,n3,n4,n5),n6) by Th51
    .=crossover(crossover(p1,p2,n3,n2,n1,n4,n5), crossover(p2,p1,n3,n2,n1,n4
  ,n5),n6) by Th51;
  hence crossover(p1,p2,n1,n2,n3,n4,n5,n6)=crossover(p1,p2,n3,n2,n1,n4,n5,n6);
  crossover(p1,p2,n1,n2,n3,n4,n5,n6) =crossover(crossover(p1,p2,n4,n2,n3,
  n1,n5), crossover(p2,p1,n1,n2,n3,n4,n5),n6) by Th51
    .=crossover(crossover(p1,p2,n4,n2,n3,n1,n5), crossover(p2,p1,n4,n2,n3,n1
  ,n5),n6) by Th51;
  hence crossover(p1,p2,n1,n2,n3,n4,n5,n6)=crossover(p1,p2,n4,n2,n3,n1,n5,n6);
  crossover(p1,p2,n1,n2,n3,n4,n5,n6) =crossover(crossover(p1,p2,n5,n2,n3,
  n4,n1), crossover(p2,p1,n1,n2,n3,n4,n5),n6) by Th51
    .=crossover(crossover(p1,p2,n5,n2,n3,n4,n1), crossover(p2,p1,n5,n2,n3,n4
  ,n1),n6) by Th51;
  hence crossover(p1,p2,n1,n2,n3,n4,n5,n6)=crossover(p1,p2,n5,n2,n3,n4,n1,n6);
  crossover(p1,p2,n1,n2,n3,n4,n5,n6) =crossover(crossover(p1,p2,n5,n2,n3,
  n4,n1), crossover(p2,p1,n1,n2,n3,n4,n5),n6) by Th51
    .=crossover(crossover(p1,p2,n5,n2,n3,n4,n1), crossover(p2,p1,n5,n2,n3,n4
  ,n1),n6) by Th51
    .=crossover(q1,q2,n1,n6)
    .=crossover(q1,q2,n6,n1) by Th13
    .=crossover(crossover(p1,p2,n6,n2,n3,n4,n5), crossover(p2,p1,n5,n2,n3,n4
  ,n6),n1) by A1,Th51
    .=crossover(crossover(p1,p2,n6,n2,n3,n4,n5), crossover(p2,p1,n6,n2,n3,n4
  ,n5),n1) by Th51;
  hence thesis;
end;
