 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 Lemma3:
(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,v1 holds v0 "/\" v1 = v1 "/\" 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,D being Element of L such that
A6:  C"/\"D <> D"/\"C;

A9: for v1,v0 holds
 ((v0"/\"v1)"/\"(v0"/\"(v0"\/"v1)))"\/"((v0"/\"v1)"/\"v0) = v0"/\"v1
proof let v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v0"\/"v1)) = v0 by A1;
  hence thesis by A1;
end;

A12: for v2,v1,v101 holds
 v101"\/"(((v101"\/"v1)"/\"(v2"\/"v101))"/\"(((v101"\/"v1)"/\"
    (v2"\/"v101))"\/"v101)) = (v101"\/"v1)"/\"(v2"\/"v101)
proof let v2,v1,v101;
  ((v101"\/"v1)"/\"(v2"\/"v101))"/\"v101 = v101 by A4;
  hence thesis by A1;
end;

A16: for v102,v1,v0 holds (v0"/\"(v102"\/"(v0"/\"v1)))"/\"(v0"/\"v1) = v0"/\"v1
proof let v102,v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v0"\/"v1)) = v0 by A1;
  hence thesis by A4;
end;

A24: for v102,v1,v0 holds
  (v1"/\"(v102"\/"(v0"/\"v1)))"/\"(v0"/\"v1) = v0"/\"v1
proof let v102,v1,v0;
  (v0"/\"v1)"\/"(v1"/\"(v0"\/"v1)) = v1 by A3;
  hence thesis by A4;
end;

A28: 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;

A32: for v101,v100 holds ((v100"/\"v101)"\/"v100)"\/"v100 = v100
proof
  now let v101,v2,v1,v100;
    ((v100"\/"v1)"/\"(v2"\/"v100))"/\"v100 = v100 by A4;
    hence ((v100"/\"v101)"\/"v100)"\/"v100 = v100 by A5;
  end;
  hence thesis;
end;

A42: 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 A28;
end;

A45: for v1,v101 holds
 (((v101"/\"v1)"\/"v101)"/\"v101)"\/"(((v101"/\"v1)"\/"v101)"/\"v101) =
    (v101"/\"v1)"\/"v101
proof let v1,v101;
  ((v101"/\"v1)"\/"v101)"\/"v101 = v101 by A32;
  hence thesis by A1;
end;

A49: for v1,v101 holds
 (v101"\/"((v101"\/"v1)"/\"v101))"\/"((v101"\/"v1)"/\"v101) =
   (v101"\/"v1)"/\"v101
proof let v1,v101;
  ((v101"\/"v1)"/\"v101)"/\"v101 = v101 by A28;
  hence thesis by A32;
end;

A60: for v0 holds v0"/\"v0 = v0 by A1,A4,A5,Lemma1;

A63: for v1,v0 holds
 ((v0"/\"v1)"\/"(v0"/\"(v0"/\"v1)))"\/"(v0"/\"(v0"/\"v1)) = v0"/\"(v0"/\"v1)
proof let v1,v0;
  (v0"/\"(v0"/\"v1))"/\"(v0"/\"v1) = v0"/\"v1 by A42;
  hence thesis by A32;
end;

A75: for v0,v1 holds v0"\/"(v1"/\"(v0"\/"v0)) = v0
proof let v0,v1;
  v0"/\"v0 = v0 by A60;
  hence thesis by A2;
end;

A83: for v0 holds v0 = v0"\/"v0 by A1,A2,A3,A4,A5,Lemma2;

A86: for v0,v1 holds v0"\/"(v1"/\"v0) = v0
proof let v0,v1;
  v0"\/"v0 = v0 by A83;
  hence thesis by A75;
end;

A88: for v1,v0 holds ((v0"/\"v1)"\/"v0)"/\"v0 = (v0"/\"v1)"\/"v0
proof let v1,v0;
  (((v0"/\"v1)"\/"v0)"/\"v0)"\/"(((v0"/\"v1)"\/"v0)"/\"v0) =
    ((v0"/\"v1)"\/"v0)"/\"v0 by A83;
  hence thesis by A45;
end;

A90: for v1,v0 holds (v0"/\"v1)"\/"(v0"/\"(v0"/\"v1)) = v0"/\"(v0"/\"v1)
proof let v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v0"/\"v1)) = v0"/\"v1 by A86;
  hence thesis by A63;
end;

A92: for v1,v0 holds v0"/\"v1 = v0"/\"(v0"/\"v1)
proof let v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v0"/\"v1)) = v0"/\"v1 by A86;
  hence thesis by A90;
end;

A95: for v1,v0 holds v0"\/"((v0"\/"v1)"/\"v0) = (v0"\/"v1)"/\"v0
proof let v1,v0;
  v0"\/"((v0"\/"v1)"/\"v0) = v0 by A86;
  hence thesis by A49;
end;

A97: for v1,v0 holds v0 = (v0"\/"v1)"/\"v0
proof let v1,v0;
  v0"\/"((v0"\/"v1)"/\"v0) = v0 by A86;
  hence thesis by A95;
end;

A101: for v101,v102 holds (v101"/\"(v102"\/"v101))"/\"v101 = v101
proof let v101,v102;
  v101"\/"v101 = v101 by A83;
  hence thesis by A4;
end;

A105: for v1,v101 holds v101"/\"((v101"/\"v1)"\/"v101) = (v101"/\"v1)"\/"v101
proof let v1,v101;
  ((v101"/\"v1)"\/"v101)"\/"v101 = v101 by A32;
  hence thesis by A97;
end;

A109: for v1,v100 holds
  (v100"/\"v1)"\/"(v100"/\"(v100"\/"(v100"/\"v1))) = v100
proof let v1,v100;
  v100"/\"(v100"/\"v1) = v100"/\"v1 by A92;
  hence thesis by A1;
end;

A113: for v1,v100 holds
 ((v100"/\"v1)"/\"(v100"/\"(v100"\/"(v100"/\"v1))))"\/"((v100"/\"
    (v100"/\"v1))"/\"v100) = v100"/\"(v100"/\"v1)
proof let v1,v100;
  v100"/\"(v100"/\"v1) = v100"/\"v1 by A92;
  hence thesis by A9;
end;

A116: for v1,v0 holds
 ((v0"/\"v1)"/\"(v0"/\"(v0"\/"(v0"/\"v1))))"\/"((v0"/\"v1)"/\"v0) =
  v0"/\"(v0"/\"v1)
proof let v1,v0;
  v0"/\"(v0"/\"v1) = v0"/\"v1 by A92;
  hence thesis by A113;
end;

A118: for v1,v0 holds
 ((v0"/\"v1)"/\"(v0"/\"(v0"\/"(v0"/\"v1))))"\/"((v0"/\"v1)"/\"v0) = v0"/\"v1
proof let v1,v0;
  v0"/\"(v0"/\"v1) = v0"/\"v1 by A92;
  hence thesis by A116;
end;

A121: for v102,v1,v101 holds
 ((v101"/\"v1)"\/"v101)"\/"((v101"/\"(v102"\/"((v101"/\"v1)"\/"v101)))"/\"
(((((v101"/\"v1)"\/"v101)"\/"v101)"/\"(v102"\/"((v101"/\"v1)"\/"v101)))"\/"(
(v101"/\"v1)"\/"v101))) =
  (((v101"/\"v1)"\/"v101)"\/"v101)"/\"(v102"\/"((v101"/\"v1)"\/"v101))
proof let v102,v1,v101;
  ((v101"/\"v1)"\/"v101)"\/"v101 = v101 by A32;
  hence thesis by A12;
end;

A124: for v2,v1,v0 holds
 ((v0"/\"v1)"\/"v0)"\/"((v0"/\"(v2"\/"((v0"/\"v1)"\/"v0)))"/\"
  ((v0"/\"(v2"\/"((v0"/\"v1)"\/"v0)))"\/"((v0"/\"v1)"\/"v0))) =
 (((v0"/\"v1)"\/"v0)"\/"v0)"/\"(v2"\/"((v0"/\"v1)"\/"v0))
proof let v2,v1,v0;
  ((v0"/\"v1)"\/"v0)"\/"v0 = v0 by A32;
  hence thesis by A121;
end;

A126: for v2,v1,v0 holds
 ((v0"/\"v1)"\/"v0)"\/"((v0"/\"(v2"\/"((v0"/\"v1)"\/"v0)))"/\"((v0"/\"
  (v2"\/"((v0"/\"v1)"\/"v0)))"\/"((v0"/\"v1)"\/"v0))) =
    v0"/\"(v2"\/"((v0"/\"v1)"\/"v0))
proof let v2,v1,v0;
  ((v0"/\"v1)"\/"v0)"\/"v0 = v0 by A32;
  hence thesis by A124;
end;

A129: for v1,v100 holds ((v100"/\"v1)"\/"v100)"/\"v100 = v100
proof let v1,v100;
  v100"/\"((v100"/\"v1)"\/"v100) = (v100"/\"v1)"\/"v100 by A105;
  hence thesis by A101;
end;

A132: for v1,v0 holds (v0"/\"v1)"\/"v0 = v0
proof let v1,v0;
  ((v0"/\"v1)"\/"v0)"/\"v0 = (v0"/\"v1)"\/"v0 by A88;
  hence thesis by A129;
end;

A135: for v1,v2,v0 holds
 (v0"/\"(v1"\/"(v0"/\"v2)))"/\"((v0"/\"v2)"\/"(v0"/\"(v1"\/"(v0"/\"v2)))) =
  ((v0"/\"(v1"\/"(v0"/\"v2)))"/\"(v0"/\"v2))"\/"(v0"/\"(v1"\/"(v0"/\"v2)))
proof let v1,v2,v0;
  (v0"/\"(v1"\/"(v0"/\"v2)))"/\"(v0"/\"v2) = v0"/\"v2 by A16;
  hence thesis by A105;
end;

A137: for v1,v2,v0 holds
 (v0"/\"(v1"\/"(v0"/\"v2)))"/\"((v0"/\"v2)"\/"(v0"/\"(v1"\/"(v0"/\"v2)))) =
  (v0"/\"v2)"\/"(v0"/\"(v1"\/"(v0"/\"v2)))
proof let v1,v2,v0;
  (v0"/\"(v1"\/"(v0"/\"v2)))"/\"(v0"/\"v2) = v0"/\"v2 by A16;
  hence thesis by A135;
end;

A139: for v2,v1,v0 holds
 v0"\/"((v0"/\"(v2"\/"((v0"/\"v1)"\/"v0)))"/\"((v0"/\"(v2"\/"
  ((v0"/\"v1)"\/"v0)))"\/"((v0"/\"v1)"\/"v0))) =
   v0"/\"(v2"\/"((v0"/\"v1)"\/"v0))
proof let v2,v1,v0;
  (v0"/\"v1)"\/"v0 = v0 by A132;
  hence thesis by A126;
end;

A142: for v1,v2,v0 holds
 (v0"\/"((v0"/\"(v1"\/"v0))"/\"((v0"/\"(v1"\/"((v0"/\"v2)"\/"v0)))"\/"
  ((v0"/\"v2)"\/"v0)))) = v0"/\"(v1"\/"((v0"/\"v2)"\/"v0))
proof let v1,v2,v0;
  (v0"/\"v2)"\/"v0 = v0 by A132;
  hence thesis by A139;
end;

A144: for v2,v0,v1 holds
 v0"\/"((v0"/\"(v1"\/"v0))"/\"((v0"/\"(v1"\/"v0))"\/"((v0"/\"v2)"\/"v0))) =
  v0"/\"(v1"\/"((v0"/\"v2)"\/"v0))
proof let v2,v0,v1;
  (v0"/\"v2)"\/"v0 = v0 by A132;
  hence thesis by A142;
end;

A146: for v1,v2,v0 holds
 v0"\/"((v0"/\"(v1"\/"v0))"/\"((v0"/\"(v1"\/"v0))"\/"v0)) =
    v0"/\"(v1"\/"((v0"/\"v2)"\/"v0))
proof let v1,v2,v0;
  (v0"/\"v2)"\/"v0 = v0 by A132;
  hence thesis by A144;
end;

A148: for v1,v2,v0 holds
 v0"\/"((v0"/\"(v1"\/"v0))"/\"v0) = v0"/\"(v1"\/"((v0"/\"v2)"\/"v0))
proof let v1,v2,v0;
  (v0"/\"(v1"\/"v0))"\/"v0 = v0 by A132;
  hence thesis by A146;
end;

A150: for v1,v2,v0 holds v0"\/"v0 = v0"/\"(v1"\/"((v0"/\"v2)"\/"v0))
proof let v1,v2,v0;
  (v0"/\"(v1"\/"v0))"/\"v0 = v0 by A101;
  hence thesis by A148;
end;

A152: for v1,v2,v0 holds v0 = v0"/\"(v1"\/"((v0"/\"v2)"\/"v0))
proof let v1,v2,v0;
  v0"\/"v0 = v0 by A83;
  hence thesis by A150;
end;

A154: for v0,v1 holds v0 = v0"/\"(v1"\/"v0)
proof
  now let v1,v2,v0;
    (v0"/\"v2)"\/"v0 = v0 by A132;
    hence v0 = v0"/\"(v1"\/"v0) by A152;
  end;
  hence thesis;
end;

A157: for v1,v2,v0 holds
 v0"/\"(v1"\/"(v0"/\"v2)) = (v0"/\"v2)"\/"(v0"/\"(v1"\/"(v0"/\"v2)))
proof let v1,v2,v0;
  (v0"/\"(v1"\/"(v0"/\"v2)))"/\"((v0"/\"v2)"\/"(v0"/\"(v1"\/"(v0"/\"v2)))) =
     v0"/\"(v1"\/"(v0"/\"v2)) by A154;
  hence thesis by A137;
end;

A161: for v1,v0 holds v0"/\"(v0"\/"(v0"/\"v1)) = v0
proof let v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v0"\/"(v0"/\"v1))) =
    v0"/\"(v0"\/"(v0"/\"v1)) by A157;
  hence thesis by A109;
end;

A163: for v1,v0 holds ((v0"/\"v1)"/\"v0)"\/"((v0"/\"v1)"/\"v0) = v0"/\"v1
proof let v1,v0;
  v0"/\"(v0"\/"(v0"/\"v1)) = v0 by A161;
  hence thesis by A118;
end;

A165: for v1,v0 holds (v0"/\"v1)"/\"v0 = v0"/\"v1
proof let v1,v0;
  ((v0"/\"v1)"/\"v0)"\/"((v0"/\"v1)"/\"v0) = (v0"/\"v1)"/\"v0 by A83;
  hence thesis by A163;
end;

A168: for v1,v100 holds v100"\/"(v100"/\"v1) = v100
proof let v1,v100;
  (v100"/\"v1)"/\"v100 = v100"/\"v1 by A165;
  hence thesis by A86;
end;

A172: for v102,v100 holds (v100"/\"v102)"/\"(v102"/\"v100) = v102"/\"v100
proof let v102,v100;
  v102"\/"(v102"/\"v100) = v102 by A168;
  hence thesis by A24;
end;

A176: for v0,v1 holds (v1"/\"v0)"/\"(v0"/\"v1) = (v0"/\"v1)"/\"(v1"/\"v0)
proof let v0,v1;
  (v0"/\"v1)"/\"(v1"/\"v0) = v1"/\"v0 by A172;
  hence thesis by A165;
end;

A179: for v0,v1 holds v1"/\"v0 = (v1"/\"v0)"/\"(v0"/\"v1)
proof let v0,v1;
  (v0"/\"v1)"/\"(v1"/\"v0) = v1"/\"v0 by A172;
  hence thesis by A176;
end;

for v1,v0 holds v0"/\"v1 = v1"/\"v0
proof let v1,v0;
  (v0"/\"v1)"/\"(v1"/\"v0) = v1"/\"v0 by A172;
  hence thesis by A179;
end;
hence thesis by A6;
end;
