
theorem Th5:
  for L being non empty RelStr, X being set, x being Element of L
holds (x is_>=_than X iff x is_>=_than X /\ the carrier of L) & (x is_<=_than X
  iff x is_<=_than X /\ the carrier of L)
proof
  let L be non empty RelStr, X be set, x be Element of L;
  set Y = X /\ the carrier of L;
  thus X is_<=_than x implies Y is_<=_than x
  proof
    assume
A1: for b being Element of L st b in X holds b <= x;
    let b be Element of L;
    assume b in Y;
    then b in X by XBOOLE_0:def 4;
    hence thesis by A1;
  end;
  thus Y is_<=_than x implies X is_<=_than x
  proof
    assume
A2: for b being Element of L st b in Y holds b <= x;
    let b be Element of L;
    assume b in X;
    then b in Y by XBOOLE_0:def 4;
    hence thesis by A2;
  end;
  thus X is_>=_than x implies Y is_>=_than x
  proof
    assume
A3: for b being Element of L st b in X holds b >= x;
    let b be Element of L;
    assume b in Y;
    then b in X by XBOOLE_0:def 4;
    hence thesis by A3;
  end;
  thus Y is_>=_than x implies X is_>=_than x
  proof
    assume
A4: for b being Element of L st b in Y holds b >= x;
    let b be Element of L;
    assume b in X;
    then b in Y by XBOOLE_0:def 4;
    hence thesis by A4;
  end;
end;
