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,p2,0,0,0,0,n5)=crossover(p1,p2,n5) & crossover(p1,p2,0,0,
0,n4,0)=crossover(p1,p2,n4) & crossover(p1,p2,0,0,n3,0,0)=crossover(p1,p2,n3) &
crossover(p1,p2,0,n2,0,0,0)=crossover(p1,p2,n2) & crossover(p1,p2,n1,0,0,0,0)=
  crossover(p1,p2,n1)
proof
A1: crossover(p1,p2,0,0,n3,0,0) =crossover(crossover(p1,p2,0,0,n3,0),
  crossover(p1,p2,n3),0) & crossover(p1,p2,0,n2,0,0,0) =crossover(crossover(p1,
  p2,0,n2,0,0), crossover(p1,p2,n2),0) by Th31;
A2: crossover(p1,p2,n1,0,0,0,0) =crossover(crossover(p1,p2,n1,0,0,0),
  crossover(p1,p2,n1),0) by Th31;
  crossover(p1,p2,0,0,0,0,n5) =crossover(p1,crossover(p2,p1,0,0,0,0),n5) &
crossover(p1,p2,0,0,0,n4,0) =crossover(crossover(p1,p2,0,0,0,n4),crossover(p1,
  p2,n4),0) by Th31,Th32;
  hence thesis by A1,A2,Th4,Th32;
end;
