
theorem Th9:
  for N being finite LATTICE holds SupMap N is one-to-one
proof
  let N be finite LATTICE;
  set f = SupMap N;
  let x, y be Element of InclPoset Ids N such that
A1: f.x = f.y;
  reconsider X = x, Y = y as Ideal of N by YELLOW_2:41;
A2: f.x = sup X & f.y = sup Y by YELLOW_2:def 3;
  X = Y
  proof
    hereby
      let a be object;
A3:   sup Y in Y by WAYBEL_3:16;
      assume
A4:   a in X;
      then reconsider b = a as Element of N;
      b <= sup Y by A1,A2,A4,YELLOW_0:17,YELLOW_4:1;
      hence a in Y by A3,WAYBEL_0:def 19;
    end;
    let a be object;
A5: sup X in X by WAYBEL_3:16;
    assume
A6: a in Y;
    then reconsider b = a as Element of N;
    b <= sup X by A1,A2,A6,YELLOW_0:17,YELLOW_4:1;
    hence thesis by A5,WAYBEL_0:def 19;
  end;
  hence thesis;
end;
