
theorem  :: Proposition 2 4L
  for R being non empty RelStr,
      X, Y being Subset of R holds
    LAp (X /\ Y) = LAp (X) /\ LAp (Y)
  proof
    let R be non empty RelStr;
    let X, Y be Subset of R;
    { x where x is Element of R : Class (the InternalRel of R,x) c= X /\ Y} =
    { x where x is Element of R : Class (the InternalRel of R,x) c= X } /\
    { x where x is Element of R : Class (the InternalRel of R,x) c= Y}
    proof
      thus
      {x where x is Element of R : Class (the InternalRel of R,x) c= X /\ Y} c=
      { x where x is Element of R : Class (the InternalRel of R,x) c= X } /\
      { x where x is Element of R : Class (the InternalRel of R,x) c= Y}
      proof
        let y be object;
        assume y in { x where x is Element of R :
        Class (the InternalRel of R,x) c= X /\ Y}; then
        consider z being Element of R such that
A1:     z = y & Class (the InternalRel of R,z) c= X /\ Y;
        X /\ Y c= X & X /\ Y c= Y by XBOOLE_1:17; then
        Class (the InternalRel of R,z) c= X &
          Class (the InternalRel of R,z) c= Y by A1; then
        z in {x where x is Element of R : Class (the InternalRel of R,x) c= X}&
        z in {x where x is Element of R : Class (the InternalRel of R,x) c= Y};
        hence thesis by A1,XBOOLE_0:def 4;
      end;
      let y be object;
      assume A2: y in { x where x is Element of R :
      Class (the InternalRel of R,x) c= X } /\
        {x where x is Element of R : Class (the InternalRel of R,x) c= Y}; then
A3:   y in { x where x is Element of R :
        Class (the InternalRel of R,x) c= X } by XBOOLE_0:def 4;
A4:   y in { x where x is Element of R :
        Class (the InternalRel of R,x) c= Y} by XBOOLE_0:def 4,A2;
      consider z being Element of R such that
A5:   z = y & Class (the InternalRel of R,z) c= X by A3;
      consider w being Element of R such that
A6:   w = y & Class (the InternalRel of R,w) c= Y by A4;
      Class (the InternalRel of R,z) /\
        Class (the InternalRel of R,z) c= X /\ Y by A5,XBOOLE_1:27,A6;
      hence thesis by A5;
    end;
    hence thesis;
  end;
