 reserve L for non empty LattStr;
 reserve v100,v102,v2,v1,v0,v3,v101 for Element of L;
 reserve L for WA_Lattice;
 reserve x,y,z,u for Element of L;
 reserve L for non empty LattStr;
 reserve v100,v101,v102,v0,v1,v2 for Element of L;
 reserve v101,v100,v2,v1,v0,v102,v103,v3 for Element of L;

theorem Lemma2:
(for v1,v0 holds (v0"/\"v1)"\/"(v0"/\"(v0"\/"v1)) = v0) &
(for v0,v1 holds (v0"/\"v0)"\/"(v1"/\"(v0"\/"v0)) = v0) &
(for v1,v0 holds (v0"/\"v1)"\/"(v1"/\"(v0"\/"v1)) = v1) &
(for v2,v1,v0 holds ((v0"\/"v1)"/\"(v2"\/"v0))"/\"v0 = v0) &
(for v2,v1,v0 holds ((v0"/\"v1)"\/"(v2"/\"v0))"\/"v0 = v0) implies
for v0 holds v0 "\/" v0 = v0
proof
assume A1: for v1,v0 holds (v0"/\"v1)"\/"(v0"/\"(v0"\/"v1)) = v0;
assume A2: for v0,v1 holds (v0"/\"v0)"\/"(v1"/\"(v0"\/"v0)) = v0;
assume A3: for v1,v0 holds (v0"/\"v1)"\/"(v1"/\"(v0"\/"v1)) = v1;
assume A4: for v2,v1,v0 holds ((v0"\/"v1)"/\"(v2"\/"v0))"/\"v0 = v0;
assume A5: for v2,v1,v0 holds ((v0"/\"v1)"\/"(v2"/\"v0))"\/"v0 = v0;
assume not thesis; then
consider C being Element of L such that
A6:  C"\/"C <> C;

A9: for v100 holds (v100"/\"v100)"\/"(v100"\/"v100) = v100
proof
  now let v2,v1,v100;
    (((v100"\/"v100)"\/"v1)"/\"(v2"\/"(v100"\/"v100)))"/\"(v100"\/"v100) =
       v100"\/"v100 by A4;
    hence (v100"/\"v100)"\/"(v100"\/"v100) = v100 by A2;
  end;
  hence thesis;
end;

A13: for v101,v100 holds ((v100"\/"v101)"/\"v100)"/\"v100 = v100
proof
  now let v101,v2,v1,v100;
    ((v100"/\"v1)"\/"(v2"/\"v100))"\/"v100 = v100 by A5;
    hence ((v100"\/"v101)"/\"v100)"/\"v100 = v100 by A4;
  end;
  hence thesis;
end;

A17: for v0 holds ((v0"/\"v0)"/\"(v0"\/"v0))"\/"((v0"/\"v0)"/\"v0) = v0"/\"v0
proof let v0;
  (v0"/\"v0)"\/"(v0"\/"v0) = v0 by A9;
  hence thesis by A1;
end;

A20: for v0 holds ((v0"/\"v0)"/\"(v0"\/"v0))"\/"((v0"\/"v0)"/\"v0) = v0"\/"v0
proof let v0;
  (v0"/\"v0)"\/"(v0"\/"v0) = v0 by A9;
  hence thesis by A3;
end;

A23: for v1,v0 holds (v0"/\"(v0"/\"v1))"/\"(v0"/\"v1) = v0"/\"v1
proof let v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v0"\/"v1)) = v0 by A1;
  hence thesis by A13;
end;

A26: for v2,v1,v101 holds
 (((v101"\/"v1)"/\"(v2"\/"v101))"/\"v101)"/\"(((v101"\/"v1)"/\"
    (v2"\/"v101))"/\"v101) = ((v101"\/"v1)"/\"(v2"\/"v101))"/\"v101
proof let v2,v1,v101;
  ((v101"\/"v1)"/\"(v2"\/"v101))"/\"v101 = v101 by A4;
  hence thesis by A23;
end;

A29: for v2,v1,v0 holds
 v0"/\"(((v0"\/"v1)"/\"(v2"\/"v0))"/\"v0) = ((v0"\/"v1)"/\"(v2"\/"v0))"/\"v0
proof let v2,v1,v0;
  ((v0"\/"v1)"/\"(v2"\/"v0))"/\"v0 = v0 by A4;
  hence thesis by A26;
end;

A31: for v2,v1,v0 holds v0"/\"v0 = ((v0"\/"v1)"/\"(v2"\/"v0))"/\"v0
proof let v2,v1,v0;
  ((v0"\/"v1)"/\"(v2"\/"v0))"/\"v0 = v0 by A4;
  hence thesis by A29;
end;

A33: for v0 holds v0"/\"v0 = v0
proof
  now let v2,v1,v0;
    ((v0"\/"v1)"/\"(v2"\/"v0))"/\"v0 = v0 by A4;
    hence v0"/\"v0 = v0 by A31;
  end;
  hence thesis;
end;

A35: for v0 holds (v0"/\"(v0"\/"v0))"\/"((v0"\/"v0)"/\"v0) = v0"\/"v0
proof let v0;
  v0"/\"v0 = v0 by A33;
  hence thesis by A20;
end;

A37: for v0 holds (v0"/\"(v0"\/"v0))"\/"((v0"/\"v0)"/\"v0) = v0"/\"v0
proof let v0;
  v0"/\"v0 = v0 by A33;
  hence thesis by A17;
end;

A39: for v0 holds (v0"/\"(v0"\/"v0))"\/"(v0"/\"v0) = v0"/\"v0
proof let v0;
  v0"/\"v0 = v0 by A33;
  hence thesis by A37;
end;

A41: for v0 holds (v0"/\"(v0"\/"v0))"\/"v0 = v0"/\"v0
proof let v0;
  v0"/\"v0 = v0 by A33;
  hence thesis by A39;
end;

A43: for v0 holds (v0"/\"(v0"\/"v0))"\/"v0 = v0
proof let v0;
  v0"/\"v0 = v0 by A33;
  hence thesis by A41;
end;

A46: for v102 holds (v102"\/"v102)"/\"v102 = v102
proof let v102;
  (v102"\/"v102)"/\"(v102"\/"v102) = v102"\/"v102 by A33;
  hence thesis by A4;
end;

A49: for v0 holds (v0"/\"(v0"\/"v0))"\/"v0 = v0"\/"v0
proof let v0;
  (v0"\/"v0)"/\"v0 = v0 by A46;
  hence thesis by A35;
end;

for v0 holds v0 = v0"\/"v0
proof let v0;
  (v0"/\"(v0"\/"v0))"\/"v0 = v0 by A43;
  hence thesis by A49;
end;
hence thesis by A6;
end;
