 reserve L for AD_Lattice;
 reserve x,y,z for Element of L;

theorem ISum:   :: Lemma 2.3. (11)
  x "\/" x = x
  proof
A8: for x,y holds (x "\/" x) "/\" y = x "/\" (y "\/" y)
    proof
      let x,y;
      set z = x;
      (x "/\" y) "\/" (x "/\" y) = x "/\" (y "\/" y) by LATTICES:def 11;
      hence thesis by DefD;
    end;
A27: x "/\" (x "\/" x) = x
    proof
      (x "\/" x) "/\" x = x "/\" (x "\/" x) by A8;
      hence thesis by DefA1;
    end;
A43:for x,y holds (x "\/" y) "/\" (y "\/" y) = y "\/" y
    proof
      let x,y;
      (x "\/" y) "/\" (y "\/" y) =
         ((x "\/" y) "/\" y) "\/" ((x "\/" y) "/\" y) by LATTICES:def 11
         .= y "\/" ((x "\/" y) "/\" y) by DefA1
         .= y "\/" y by DefA1;
      hence thesis;
    end;
A49:(x "\/" x) "\/" (x "\/" x) = x "\/" x
    proof
      (x "\/" x) "/\" (x "\/" x)
       = ((x "\/" x) "/\" x) "\/" ((x "\/" x) "/\" x) by LATTICES:def 11
      .= x "\/" ((x "\/" x) "/\" x) by DefA1
      .= x "\/" x by DefA1;
      hence thesis by ROBBINS3:def 3;
    end;
    set R = x "\/" x;
    x "\/" x = (x "\/" x) "/\" (x "\/" x) by A43
            .= x by A27,A49,A8;
    hence thesis;
  end;
