 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;
reserve L for with_zero GAD_Lattice,
        x,y for Element of L;

theorem Thx3:
  for L be non empty LattStr, S be non empty SubLattStr of L,
      x1,x2 be Element of L, y1,y2 be Element of S
  st x1 = y1 & x2 = y2 holds x1 "\/" x2 = y1 "\/" y2
  proof
    let L be non empty LattStr;
    let S be non empty SubLattStr of L;
    let x1,x2 be Element of L;
    let y1,y2 be Element of S;
    assume
Z4: x1 = y1 & x2 = y2;
Z5: the L_join of S = (the L_join of L)||the carrier of S by Defx1
    .= (the L_join of L)|[:the carrier of S,the carrier of S:];
    [y1,y2] in [:the carrier of S,the carrier of S:] by ZFMISC_1:def 2;
    hence x1 "\/" x2 = y1 "\/" y2 by Z4,Z5,FUNCT_1:49;
  end;
