reserve n,k for Element of NAT;

theorem Th3:
  for L be finite LATTICE for C be Chain of L for x,y be Element of
  L holds x in C & y in C implies ( x < y iff height(x) < height(y) )
proof
  let L be finite LATTICE;
  let C be Chain of L;
  let x,y be Element of L;
  assume
A1: x in C & y in C;
  height(x) < height(y) implies x < y
  proof
    assume
A2: height(x) < height(y);
    per cases by A1,ORDERS_2:11;
    suppose
      x <= y;
      hence thesis by A2,ORDERS_2:def 6;
    end;
    suppose
      y <= x;
      then x=y or y < x by ORDERS_2:def 6;
      hence thesis by A2,Th2;
    end;
  end;
  hence x < y iff height(x) < height(y) by Th2;
end;
