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,p2,0,0,0,n4,n5)=crossover(p2,p1,n4,n5) & crossover(p1,p2,
0,0,n3,0,n5)=crossover(p2,p1,n3,n5) & crossover(p1,p2,0,0,n3,n4,0)=crossover(p2
,p1,n3,n4) & crossover(p1,p2,0,n2,0,0,n5)=crossover(p2,p1,n2,n5) & crossover(p1
  ,p2,0,n2,0,n4,0)=crossover(p2,p1,n2,n4) & crossover(p1,p2,0,n2,n3,0,0)=
crossover(p2,p1,n2,n3) & crossover(p1,p2,n1,0,0,0,n5)=crossover(p2,p1,n1,n5) &
crossover(p1,p2,n1,0,0,n4,0)=crossover(p2,p1,n1,n4) & crossover(p1,p2,n1,0,n3,0
,0)=crossover(p2,p1,n1,n3) & crossover(p1,p2,n1,n2,0,0,0)=crossover(p2,p1,n1,n2
  )
proof
A1: crossover(p1,p2,0,0,n3,n4,0) =crossover(crossover(p1,p2,0,0,n3,n4),
crossover( p2,p1,n3,n4),0) & crossover(p1,p2,0,n2,0,n4,0) =crossover(crossover(
  p1,p2,0,n2,0,n4),crossover(p2,p1,n2,n4),0) by Th30;
A2: crossover(p1,p2,0,n2,n3,0,0) =crossover(crossover(p1,p2,0,n2,n3,0),
crossover( p2,p1,n2,n3),0) & crossover(p1,p2,n1,0,0,n4,0) =crossover(crossover(
  p1,p2,n1,0 ,0,n4),crossover(p2,p1,n1,n4),0) by Th30;
A3: crossover(p1,p2,0,0,n3,0,n5) =crossover(crossover(p2,p1,n3),crossover(p2
  ,p1,0,0,n3,0),n5) by Th31
    .=crossover(crossover(p2,p1,n3),crossover(p1,p2,n3),n5) by Th31;
A4: crossover(p1,p2,n1,0,n3,0,0) =crossover(crossover(p1,p2,n1,0,n3,0),
crossover( p2,p1,n1,n3),0) & crossover(p1,p2,n1,n2,0,0,0) =crossover(crossover(
  p1,p2,n1, n2,0,0),crossover(p2,p1,n1,n2),0) by Th30;
A5: crossover(p1,p2,n1,0,0,0,n5) =crossover(crossover(p2,p1,n1),crossover(
  p2,p1,n1,0,0,0),n5) by Th31
    .=crossover(crossover(p2,p1,n1),crossover(p1,p2,n1),n5) by Th31;
A6: crossover(p1,p2,0,n2,0,0,n5) =crossover(crossover(p2,p1,n2),crossover(p2
  ,p1,0,n2,0,0),n5) by Th31
    .=crossover(crossover(p2,p1,n2),crossover(p1,p2,n2),n5) by Th31;
  crossover(p1,p2,0,0,0,n4,n5) =crossover(crossover(p2,p1,n4),crossover(p2
  ,p1,0,0,0,n4),n5) by Th31
    .=crossover(crossover(p2,p1,n4),crossover(p1,p2,n4),n5) by Th31;
  hence thesis by A3,A6,A1,A5,A2,A4,Th4;
end;
