reserve n,k for Element of NAT;

theorem Th5:
  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;
A2: height(x) <= height(y) implies x<=y
  proof
    assume height(x) <= height(y);
    then height(x) < height(y) or height(x) = height(y) by XXREAL_0:1;
    then x<y or height(x) = height(y) by A1,Th3;
    hence thesis by A1,Th4,ORDERS_2:def 6;
  end;
  x <= y implies height(x) <= height(y)
  proof
    assume x<=y;
    then x<y or x=y by ORDERS_2:def 6;
    hence thesis by A1,Th3;
  end;
  hence x <= y iff height(x) <= height(y) by A2;
end;
