 reserve L for non empty LattStr;
 reserve v100,v102,v2,v1,v0,v3,v101 for Element of L;

theorem Lemma4:
  (for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
  (for v0 holds v0"\/"v0 = v0) &
  (for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
  (for v2,v1,v0 holds ((v0"/\"v1)"\/"(v2"/\"v1))"\/"v1 = v1) implies
    for v1,v2 holds v1"\/"(v1"/\"v2) = v1
proof
  assume A1: for v1,v0 holds v0"/\"v1 = v1"/\"v0;
  assume A2: for v0 holds v0"\/"v0 = v0;
  assume A3: for v1,v0 holds v0"\/"v1 = v1"\/"v0;
  assume A4: for v2,v1,v0 holds ((v0"/\"v1)"\/"(v2"/\"v1))"\/"v1 = v1;
A6: for v2,v1,v0 holds v1"\/"((v0"/\"v1)"\/"(v2"/\"v1)) = v1
  proof let v2,v1,v0;
    ((v0"/\"v1)"\/"(v2"/\"v1))"\/"v1 = v1"\/"((v0"/\"v1)"\/"(v2"/\"v1)) by A3;
    hence thesis by A4;
  end;
A11: for v100,v102 holds v100"\/"(v102"/\"v100) = v100
  proof let v100,v102;
    (v102"/\"v100)"\/"(v102"/\"v100) = v102"/\"v100 by A2;
    hence thesis by A6;
  end;
  for v1,v0 holds v0"\/"(v0"/\"v1) = v0
  proof let v1,v0;
    v1"/\"v0 = v0"/\"v1 by A1;
    hence thesis by A11;
  end;
  hence thesis;
end;
