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,n3,n4,n5)=crossover(p1,p2,n3,n4,n5) & crossover(p1
  ,p2,0,n2,0,n4,n5)=crossover(p1,p2,n2,n4,n5) & crossover(p1,p2,0,n2,n3,0,n5)=
crossover(p1,p2,n2,n3,n5) & crossover(p1,p2,0,n2,n3,n4,0)=crossover(p1,p2,n2,n3
,n4) & crossover(p1,p2,n1,0,0,n4,n5)=crossover(p1,p2,n1,n4,n5) & crossover(p1,
  p2,n1,0,n3,0,n5)=crossover(p1,p2,n1,n3,n5) & crossover(p1,p2,n1,0,n3,n4,0)=
crossover(p1,p2,n1,n3,n4) & crossover(p1,p2,n1,n2,0,0,n5)=crossover(p1,p2,n1,n2
,n5) & crossover(p1,p2,n1,n2,0,n4,0)=crossover(p1,p2,n1,n2,n4) & crossover(p1,
  p2,n1,n2,n3,0,0)=crossover(p1,p2,n1,n2,n3)
proof
A1: crossover(p1,p2,0,n2,n3,n4,0) =crossover(crossover(p1,p2,0,n2,n3,n4),
  crossover(p1,p2,n2,n3,n4),0) & crossover(p1,p2,n1,0,n3,n4,0) =crossover(
  crossover(p1,p2,n1,0,n3,n4),crossover(p1,p2,n1,n3,n4),0) by Th29;
A2: crossover(p1,p2,n1,n2,0,n4,0) =crossover(crossover(p1,p2,n1,n2,0,n4),
  crossover(p1,p2,n1,n2,n4),0) & crossover(p1,p2,n1,n2,n3,0,0) =crossover(
  crossover(p1,p2,n1,n2,n3,0),crossover(p1,p2,n1,n2,n3),0) by Th29;
A3: crossover(p1,p2,n1,0,n3,0,n5) =crossover(crossover(p1,p2,n1,n3),
  crossover(p2,p1,n1,0,n3,0),n5) by Th30
    .=crossover(crossover(p1,p2,n1,n3),crossover(p2,p1,n1,n3),n5) by Th30;
A4: crossover(p1,p2,n1,0,0,n4,n5) =crossover(crossover(p1,p2,n1,n4),
  crossover(p2,p1,n1,0,0,n4),n5) by Th30
    .=crossover(crossover(p1,p2,n1,n4),crossover(p2,p1,n1,n4),n5) by Th30;
A5: crossover(p1,p2,n1,n2,0,0,n5) =crossover(crossover(p1,p2,n1,n2),
  crossover(p2,p1,n1,n2,0,0),n5) by Th30
    .=crossover(crossover(p1,p2,n1,n2),crossover(p2,p1,n1,n2),n5) by Th30;
A6: crossover(p1,p2,0,n2,n3,0,n5) =crossover(crossover(p1,p2,n2,n3),
  crossover(p2,p1,0,n2,n3,0),n5) by Th30
    .=crossover(crossover(p1,p2,n2,n3),crossover(p2,p1,n2,n3),n5) by Th30;
A7: crossover(p1,p2,0,n2,0,n4,n5) =crossover(crossover(p1,p2,n2,n4),
  crossover(p2,p1,0,n2,0,n4),n5) by Th30
    .=crossover(crossover(p1,p2,n2,n4),crossover(p2,p1,n2,n4),n5) by Th30;
  crossover(p1,p2,0,0,n3,n4,n5) =crossover(crossover(p1,p2,n3,n4),
  crossover(p2,p1,0,0,n3,n4),n5) by Th30
    .=crossover(crossover(p1,p2,n3,n4),crossover(p2,p1,n3,n4),n5) by Th30;
  hence thesis by A7,A6,A4,A3,A1,A5,A2,Th4;
end;
