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 Th34:
  (n1 >= len p1 & n2 >= len p1 implies crossover(p1,p2,n1,n2,n3,n4
) = crossover(p1,p2,n3,n4)) & (n1 >= len p1 & n3 >= len p1 implies crossover(p1
  ,p2,n1,n2,n3,n4) = crossover(p1,p2,n2,n4)) & (n1 >= len p1 & n4 >= len p1
implies crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n2,n3)) & (n2 >= len p1
& n3 >= len p1 implies crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n1,n4)) &
(n2 >= len p1 & n4 >= len p1 implies crossover(p1,p2,n1,n2,n3,n4) = crossover(
p1,p2,n1,n3)) & (n3 >= len p1 & n4 >= len p1 implies crossover(p1,p2,n1,n2,n3,
  n4) = crossover(p1,p2,n1,n2))
proof
A1: n1 >= len p1 & n3 >= len p1 implies crossover(p1,p2,n1,n2,n3,n4) =
  crossover(p1,p2,n2,n4)
  proof
    assume that
A2: n1 >= len p1 and
A3: n3 >= len p1;
    crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n2,n3,n4) by A2,Th33;
    hence thesis by A3,Th19;
  end;
A4: n1 >= len p1 & n4 >= len p1 implies crossover(p1,p2,n1,n2,n3,n4) =
  crossover(p1,p2,n2,n3)
  proof
    assume that
A5: n1 >= len p1 and
A6: n4 >= len p1;
    crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n2,n3,n4) by A5,Th33;
    hence thesis by A6,Th20;
  end;
A7: n3 >= len p1 & n4 >= len p1 implies crossover(p1,p2,n1,n2,n3,n4) =
  crossover(p1,p2,n1,n2)
  proof
    assume that
A8: n3 >= len p1 and
A9: n4 >= len p1;
    crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n1,n2,n4) by A8,Th33;
    hence thesis by A9,Th20;
  end;
A10: n2 >= len p1 & n4 >= len p1 implies crossover(p1,p2,n1,n2,n3,n4) =
  crossover(p1,p2,n1,n3)
  proof
    assume that
A11: n2 >= len p1 and
A12: n4 >= len p1;
    crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n1,n3,n4) by A11,Th33;
    hence thesis by A12,Th20;
  end;
A13: n2 >= len p1 & n3 >= len p1 implies crossover(p1,p2,n1,n2,n3,n4) =
  crossover(p1,p2,n1,n4)
  proof
    assume that
A14: n2 >= len p1 and
A15: n3 >= len p1;
    crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n1,n3,n4) by A14,Th33;
    hence thesis by A15,Th19;
  end;
  n1 >= len p1 & n2 >= len p1 implies crossover(p1,p2,n1,n2,n3,n4) =
  crossover(p1,p2,n3,n4)
  proof
    assume that
A16: n1 >= len p1 and
A17: n2 >= len p1;
    crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n2,n3,n4) by A16,Th33;
    hence thesis by A17,Th18;
  end;
  hence thesis by A1,A4,A13,A10,A7;
end;
