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

theorem Th381i4: :: Theorem 3.8. 1) <=> 4)
  x "\/" y = y "\/" x iff ex_lub_of x,y & x "\/" y = lub (x,y)
  proof
    hereby
      assume x "\/" y = y "\/" x; then
      ex c being Element of L st x [= c & y [= c by Th3823,LemX3;
      hence ex_lub_of x,y & x "\/" y = lub (x,y) by Th3834;
    end;
    assume ex_lub_of x,y & x "\/" y = lub (x,y);
    hence thesis by Th3841;
  end;
