reserve Y for non empty set;
reserve B for Subset of Y;

theorem
  for a,b,c being Function of Y,BOOLEAN holds ( a '<' b & b '<' a
  implies a=b)& ( a '<' b & b '<' c implies a '<' c)
proof
  let a,b,c be Function of Y,BOOLEAN;
A1: for a,b,c being Function of Y,BOOLEAN holds a '<' b & b '<' a
  implies a = b
  proof
    let a,b,c be Function of Y,BOOLEAN;
    assume
A2: a '<' b & b '<' a;
      let x be Element of Y;
      a.x = FALSE & b.x = FALSE or a.x = FALSE & b.x = TRUE or a.x = TRUE
      & b.x = FALSE or a.x = TRUE & b.x = TRUE by XBOOLEAN:def 3;
      hence thesis by A2;
  end;
  for a,b,c being Function of Y,BOOLEAN holds a '<' b & b '<' c
  implies a '<' c
  proof
    let a,b,c be Function of Y,BOOLEAN;
    assume that
A3: a '<' b and
A4: b '<' c;
    for x being Element of Y st a.x= TRUE holds c.x=TRUE
    proof
      let x be Element of Y;
      b.x = TRUE implies c.x = TRUE by A4;
      hence thesis by A3;
    end;
    hence thesis;
  end;
  hence thesis by A1;
end;
