reserve n,k for Element of NAT;

theorem Th4:
  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;
  thus x = y implies height(x) = height(y);
  height(x) = height(y) implies x=y
  proof
    assume that
A2: height(x) = height(y) and
A3: x<>y;
A4: x<=y or y<=x by A1,ORDERS_2:11;
    height(x)<>height(y)
    proof
      per cases by A3,A4,ORDERS_2:def 6;
      suppose
        x < y;
        hence thesis by A1,Th3;
      end;
      suppose
        y < x;
        hence thesis by A1,Th3;
      end;
    end;
    hence contradiction by A2;
  end;
  hence thesis;
end;
