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 Th41:
  crossover(p1,p2,0,n2,n3,n4,n5)=crossover(p2,p1,n2,n3,n4,n5) &
crossover(p1,p2,n1,0,n3,n4,n5) = crossover(p2,p1,n1,n3,n4,n5) & crossover(p1,p2
,n1,n2,0,n4,n5) = crossover(p2,p1,n1,n2,n4,n5) & crossover(p1,p2,n1,n2,n3,0,n5)
= crossover(p2,p1,n1,n2,n3,n5) & crossover(p1,p2,n1,n2,n3,n4,0) = crossover(p2,
  p1,n1,n2,n3,n4)
proof
A1: crossover(p1,p2,n1,n2,n3,n4,0) =crossover(crossover(p1,p2,n1,n2,n3,n4),
  crossover(p2,p1,n1,n2,n3,n4),0);
A2: crossover(p1,p2,n1,0,n3,n4,n5) =crossover(crossover(p2,p1,n1,n3,n4),
  crossover(p2,p1,n1,0,n3,n4),n5) by Th29
    .=crossover(crossover(p2,p1,n1,n3,n4),crossover(p1,p2,n1,n3,n4),n5) by Th29
;
A3: crossover(p1,p2,n1,n2,n3,0,n5) =crossover(crossover(p2,p1,n1,n2,n3),
  crossover(p2,p1,n1,n2,n3,0),n5) by Th29
    .=crossover(crossover(p2,p1,n1,n2,n3),crossover(p1,p2,n1,n2,n3),n5) by Th29
;
A4: crossover(p1,p2,n1,n2,0,n4,n5) =crossover(crossover(p2,p1,n1,n2,n4),
  crossover(p2,p1,n1,n2,0,n4),n5) by Th29
    .=crossover(crossover(p2,p1,n1,n2,n4),crossover(p1,p2,n1,n2,n4),n5) by Th29
;
  crossover(p1,p2,0,n2,n3,n4,n5) =crossover(crossover(p2,p1,n2,n3,n4),
  crossover(p2,p1,0,n2,n3,n4),n5) by Th29
    .=crossover(crossover(p2,p1,n2,n3,n4),crossover(p1,p2,n2,n3,n4),n5) by Th29
;
  hence thesis by A2,A4,A3,A1,Th4;
end;
