reserve L for non empty LattStr;
reserve v3,v101,v100,v102,v103,v2,v1,v0 for Element of L;

theorem ThQLT6:
(for v0 holds v0"/\"v0 = v0) &
(for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)) &
(for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
(for v0,v2,v1 holds
  (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
(for v0 holds v0"\/"v0 = v0) &
(for v2,v1,v0 holds
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
(for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
(for v0,v2,v1 holds
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
(for v2,v1,v0 holds
  ((v0"\/"v1)"/\"v2)"\/"v1 = ((v2"\/"v1)"/\"v0)"\/"v1)
implies
for v1,v2,v3 holds
  (v1"/\"v2)"\/"(v1"/\"v3) = v1"/\"(v2"\/"(v1"/\"v3))
proof
assume A2: for v0 holds v0"/\"v0 = v0;
assume A3: for v2,v1,v0 holds
  (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2);
assume A4: for v1,v0 holds v0"/\"v1 = v1"/\"v0;
assume A5: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2);
assume A6: for v0 holds v0"\/"v0 = v0;
assume A7: for v2,v1,v0 holds
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2);
assume A8: for v1,v0 holds v0"\/"v1 = v1"\/"v0;
assume A9: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2);

A11: for v0,v2,v1 holds
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) =
    (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) by A4;
  hence thesis by A9;
end;

assume A12: for v2,v1,v0 holds
 ((v0"\/"v1)"/\"v2)"\/"v1 = ((v2"\/"v1)"/\"v0)"\/"v1;

A14: for v2,v1,v0 holds
  v1"\/"((v0"\/"v1)"/\"v2) = ((v2"\/"v1)"/\"v0)"\/"v1
proof let v2,v1,v0;
  ((v0"\/"v1)"/\"v2)"\/"v1 = v1"\/"((v0"\/"v1)"/\"v2) by A8;
  hence thesis by A12;
end;

A17: for v2,v0,v1 holds
  v0"\/"((v1"\/"v0)"/\"v2) = v0"\/"((v2"\/"v0)"/\"v1)
proof let v2,v0,v1;
  ((v2"\/"v0)"/\"v1)"\/"v0 = v0"\/"((v2"\/"v0)"/\"v1) by A8;
  hence thesis by A14;
end;

A21: for v0,v2,v1 holds
  (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) =
    (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) by A8;
  hence thesis by A5;
end;

A24: for v102,v101 holds v101"/\"v102 = v101"/\"(v101"/\"v102)
proof let v102,v101;
  v101"/\"v101 = v101 by A2;
  hence thesis by A3;
end;

A28: for v2,v0,v1 holds (v1"/\"v0)"/\"v2 = v0"/\"(v1"/\"v2)
proof let v2,v0,v1;
  v0"/\"v1 = v1"/\"v0 by A4;
  hence thesis by A3;
end;

A31: for v0,v2,v1 holds v0"/\"(v1"/\"v2) = v1"/\"(v0"/\"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2) by A3;
  hence thesis by A28;
end;

A34: for v102,v100 holds (v100"\/"v102)"\/"v102 = v100"\/"v102
proof let v102,v100;
  v102"\/"v102 = v102 by A6;
  hence thesis by A7;
end;

A37: for v1,v0 holds
  v1"\/"(v0"\/"v1) = v0"\/"v1
proof let v1,v0;
  (v0"\/"v1)"\/"v1 = v1"\/"(v0"\/"v1) by A8;
  hence thesis by A34;
end;

A40: for v0,v1,v2 holds
  (v0"\/"v1)"/\"(v0"\/"(v2"/\"v1)) = v0"\/"(v1"/\"v2)
proof let v0,v1,v2;
  v1"/\"v2 = v2"/\"v1 by A4;
  hence thesis by A11;
end;

A43: for v100,v101 holds
 v100"\/"(v101"\/"v100) =
    v100"\/"(((v101"\/"v100)"\/"v100)"/\"v101)
proof let v100,v101;
  (v101"\/"v100)"/\"(v101"\/"v100) = v101"\/"v100 by A2;
  hence thesis by A17;
end;

A46: for v0,v1 holds v1"\/"v0 = v0"\/"(((v1"\/"v0)"\/"v0)"/\"v1)
proof let v0,v1;
  v0"\/"(v1"\/"v0) = v1"\/"v0 by A37;
  hence thesis by A43;
end;

A49: for v1,v0 holds
  v0"\/"v1 = v1"\/"((v1"\/"(v0"\/"v1))"/\"v0)
proof let v1,v0;
  (v0"\/"v1)"\/"v1 = v1"\/"(v0"\/"v1) by A8;
  hence thesis by A46;
end;

A51: for v1,v0 holds
  v0"\/"v1 = v1"\/"((v0"\/"v1)"/\"v0)
proof let v1,v0;
  v1"\/"(v0"\/"v1) = v0"\/"v1 by A37;
  hence thesis by A49;
end;

A53: for v1,v0 holds
  v0"\/"v1 = v1"\/"(v0"/\"(v0"\/"v1))
proof let v1,v0;
  (v0"\/"v1)"/\"v0 = v0"/\"(v0"\/"v1) by A4;
  hence thesis by A51;
end;

A57: for v2,v0,v1 holds
  v0"\/"(v2"/\"(v1"\/"v0)) = v0"\/"((v2"\/"v0)"/\"v1)
proof let v2,v0,v1;
  (v1"\/"v0)"/\"v2 = v2"/\"(v1"\/"v0) by A4;
  hence thesis by A17;
end;

A62: for v102,v100 holds
 v100"\/"(v100"/\"v102) = v100"\/"((v102"\/"v100)"/\"v100)
proof let v102,v100;
  v100"\/"v100 = v100 by A6;
  hence thesis by A17;
end;

A65: for v1,v0 holds v0"\/"(v0"/\"v1) = v0"\/"(v0"/\"(v1"\/"v0))
proof let v1,v0;
  (v1"\/"v0)"/\"v0 = v0"/\"(v1"\/"v0) by A4;
  hence thesis by A62;
end;

A69: for v102,v2,v100,v1 holds
 (v100"\/"((v2"\/"v100)"/\"v1))"/\"
   (v100"\/"(((v1"\/"v100)"/\"v2)"/\"v102)) =
    v100"\/"(((v1"\/"v100)"/\"v2)"/\"v102)
proof let v102,v2,v100,v1;
  v100"\/"((v1"\/"v100)"/\"v2) =
    v100"\/"((v2"\/"v100)"/\"v1) by A17;
  hence thesis by A11;
end;

A72: for v3,v1,v0,v2 holds
 (v0"\/"(v1"/\"(v2"\/"v0)))"/\"
   (v0"\/"(((v2"\/"v0)"/\"v1)"/\"v3)) =
     v0"\/"(((v2"\/"v0)"/\"v1)"/\"v3)
proof let v3,v1,v0,v2;
  v0"\/"((v1"\/"v0)"/\"v2) = v0"\/"(v1"/\"(v2"\/"v0)) by A57;
  hence thesis by A69;
end;

A74: for v1,v3,v0,v2 holds
 (v0"\/"(v1"/\"(v2"\/"v0)))"/\"(v0"\/"((v2"\/"v0)"/\"(v1"/\"v3))) =
    v0"\/"(((v2"\/"v0)"/\"v1)"/\"v3)
proof let v1,v3,v0,v2;
  ((v2"\/"v0)"/\"v1)"/\"v3 = (v2"\/"v0)"/\"(v1"/\"v3) by A3;
  hence thesis by A72;
end;

A76: for v2,v0,v3,v1 holds
 (v0"\/"(v1"/\"(v2"\/"v0)))"/\"(v0"\/"(v2"/\"((v1"/\"v3)"\/"v0))) =
   v0"\/"(((v2"\/"v0)"/\"v1)"/\"v3)
proof let v2,v0,v3,v1;
  v0"\/"((v2"\/"v0)"/\"(v1"/\"v3)) =
    v0"\/"(v2"/\"((v1"/\"v3)"\/"v0)) by A57;
  hence thesis by A74;
end;

A78: for v2,v0,v3,v1 holds
  (v0"\/"(v1"/\"(v2"\/"v0)))"/\"(v0"\/"(v2"/\"((v1"/\"v3)"\/"v0))) =
     v0"\/"((v2"\/"v0)"/\"(v1"/\"v3))
proof let v2,v0,v3,v1;
  ((v2"\/"v0)"/\"v1)"/\"v3 = (v2"\/"v0)"/\"(v1"/\"v3) by A3;
  hence thesis by A76;
end;

A80: for v2,v0,v3,v1 holds
 (v0"\/"(v1"/\"(v2"\/"v0)))"/\"(v0"\/"(v2"/\"((v1"/\"v3)"\/"v0))) =
    v0"\/"(v2"/\"((v1"/\"v3)"\/"v0))
proof let v2,v0,v3,v1;
  v0"\/"((v2"\/"v0)"/\"(v1"/\"v3)) = v0"\/"(v2"/\"((v1"/\"v3)"\/"v0)) by A57;
  hence thesis by A78;
end;

A83: for v102,v101 holds
  v101"\/"(v101"/\"(v101"\/"v102)) = v101"/\"(v101"\/"v102)
proof let v102,v101;
  v101"/\"v101 = v101 by A2;
  hence thesis by A21;
end;

A87: for v100,v101,v1 holds
 (v100"/\"v101)"\/"(v100"/\"(v1"\/"v101)) =
    v100"/\"(v101"\/"(v1"\/"v101))
proof let v100,v101,v1;
  v101"\/"(v1"\/"v101) = v1"\/"v101 by A37;
  hence thesis by A21;
end;

A90: for v0,v1,v2 holds
  (v0"/\"v1)"\/"(v0"/\"(v2"\/"v1)) = v0"/\"(v2"\/"v1)
proof let v0,v1,v2;
  v1"\/"(v2"\/"v1) = v2"\/"v1 by A37;
  hence thesis by A87;
end;

A93: for v101,v1 holds
 (v1"\/"v101)"\/"(v101"/\"(v1"\/"v101)) = v101"\/"(v1"\/"v101)
proof let v101,v1;
  v101"\/"(v1"\/"v101) = v1"\/"v101 by A37;
  hence thesis by A53;
end;

A96: for v1,v0 holds
  v0"\/"(v1"\/"(v1"/\"(v0"\/"v1))) = v1"\/"(v0"\/"v1)
proof let v1,v0;
  (v0"\/"v1)"\/"(v1"/\"(v0"\/"v1)) =
    v0"\/"(v1"\/"(v1"/\"(v0"\/"v1))) by A7;
  hence thesis by A93;
end;

A98: for v0,v1 holds
  v0"\/"(v1"\/"(v1"/\"v0)) = v1"\/"(v0"\/"v1)
proof let v0,v1;
  v1"\/"(v1"/\"(v0"\/"v1)) = v1"\/"(v1"/\"v0) by A65;
  hence thesis by A96;
end;

A100: for v0,v1 holds v0"\/"(v1"\/"(v1"/\"v0)) = v0"\/"v1
proof let v0,v1;
  v1"\/"(v0"\/"v1) = v0"\/"v1 by A37;
  hence thesis by A98;
end;

A103: for v0,v100,v1 holds
 v100"\/"((v0"/\"v1)"\/"(v0"/\"(v1"/\"v100))) =
   v100"\/"(v0"/\"v1)
proof let v0,v100,v1;
  (v0"/\"v1)"/\"v100 = v0"/\"(v1"/\"v100) by A3;
  hence thesis by A100;
end;

A106: for v1,v0 holds
  v0"\/"(v0"/\"(v0"\/"v1)) = v0"\/"(v0"/\"v1)
proof let v1,v0;
  v1"\/"v0 = v0"\/"v1 by A8;
  hence thesis by A65;
end;

A108: for v1,v0 holds
  v0"/\"(v0"\/"v1) = v0"\/"(v0"/\"v1)
proof let v1,v0;
  v0"\/"(v0"/\"(v0"\/"v1)) = v0"/\"(v0"\/"v1) by A83;
  hence thesis by A106;
end;

A111: for v100,v1,v101 holds
  v100"/\"(v101"\/"(v101"/\"v1)) = v101"/\"(v100"/\"(v101"\/"v1))
proof let v100,v1,v101;
  v101"/\"(v101"\/"v1) = v101"\/"(v101"/\"v1) by A108;
  hence thesis by A31;
end;

A115: for v1,v100,v101 holds
 (v100"\/"v101)"/\"(v100"\/"(v1"/\"(v101"\/"v100))) =
    v100"\/"(v101"/\"(v1"\/"v100))
proof let v1,v100,v101;
  v100"\/"((v1"\/"v100)"/\"v101) =
    v100"\/"(v1"/\"(v101"\/"v100)) by A57;
  hence thesis by A40;
end;

A119: for v102,v1,v100 holds
 (v100"/\"v1)"\/"(v100"/\"(v102"\/"(v100"/\"v1))) =
    v100"/\"(v102"\/"(v100"/\"v1))
proof let v102,v1,v100;
  v100"/\"(v100"/\"v1) = v100"/\"v1 by A24;
  hence thesis by A90;
end;

A123: for v100,v103,v101 holds
 (v100"\/"(v101"/\"((v101"/\"v103)"\/"v100)))"/\"
   (v100"\/"((v101"/\"v103)"\/"
    ((v101"/\"v103)"/\"v100))) =
      v100"\/"((v101"/\"v103)"/\"((v101"/\"v103)"\/"v100))
proof let v100,v103,v101;
  (v101"/\"v103)"/\"((v101"/\"v103)"\/"v100) =
    (v101"/\"v103)"\/"((v101"/\"v103)"/\"v100) by A108;
  hence thesis by A80;
end;

A126: for v0,v2,v1 holds
  (v0"\/"(v1"/\"((v1"/\"v2)"\/"v0)))"/\"
    (v0"\/"((v1"/\"v2)"\/"(v1"/\"(v2"/\"v0)))) =
      v0"\/"((v1"/\"v2)"/\"((v1"/\"v2)"\/"v0))
proof let v0,v2,v1;
  (v1"/\"v2)"/\"v0 = v1"/\"(v2"/\"v0) by A3;
  hence thesis by A123;
end;

A128: for v0,v2,v1 holds
 (v0"\/"(v1"/\"((v1"/\"v2)"\/"v0)))"/\"(v0"\/"(v1"/\"v2)) =
    v0"\/"((v1"/\"v2)"/\"((v1"/\"v2)"\/"v0))
proof let v0,v2,v1;
  v0"\/"((v1"/\"v2)"\/"(v1"/\"(v2"/\"v0))) = v0"\/"(v1"/\"v2) by A103;
  hence thesis by A126;
end;

A130: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v0"\/"(v1"/\"((v1"/\"v2)"\/"v0))) =
    v0"\/"((v1"/\"v2)"/\"((v1"/\"v2)"\/"v0))
proof let v0,v2,v1;
  (v0"\/"(v1"/\"((v1"/\"v2)"\/"v0)))"/\"(v0"\/"(v1"/\"v2)) =
    (v0"\/"(v1"/\"v2))"/\"(v0"\/"(v1"/\"((v1"/\"v2)"\/"v0))) by A4;
  hence thesis by A128;
end;

A132: for v0,v2,v1 holds
  v0"\/"((v1"/\"v2)"/\"(v1"\/"v0)) =
    v0"\/"((v1"/\"v2)"/\"((v1"/\"v2)"\/"v0))
proof let v0,v2,v1;
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"(v1"/\"((v1"/\"v2)"\/"v0))) =
    v0"\/"((v1"/\"v2)"/\"(v1"\/"v0)) by A115;
  hence thesis by A130;
end;

A134: for v2,v0,v1 holds
  v0"\/"(v1"/\"(v2"/\"(v1"\/"v0))) =
    v0"\/"((v1"/\"v2)"/\"((v1"/\"v2)"\/"v0))
proof let v2,v0,v1;
  (v1"/\"v2)"/\"(v1"\/"v0) = v1"/\"(v2"/\"(v1"\/"v0)) by A3;
  hence thesis by A132;
end;

A136: for v2,v0,v1 holds
  v0"\/"(v1"/\"(v2"/\"(v1"\/"v0))) =
    v0"\/"((v1"/\"v2)"\/"((v1"/\"v2)"/\"v0))
proof let v2,v0,v1;
  (v1"/\"v2)"/\"((v1"/\"v2)"\/"v0) =
    (v1"/\"v2)"\/"((v1"/\"v2)"/\"v0) by A108;
  hence thesis by A134;
end;

A138: for v2,v0,v1 holds
  v0"\/"(v1"/\"(v2"/\"(v1"\/"v0))) =
    v0"\/"((v1"/\"v2)"\/"(v1"/\"(v2"/\"v0)))
proof let v2,v0,v1;
  (v1"/\"v2)"/\"v0 = v1"/\"(v2"/\"v0) by A3;
  hence thesis by A136;
end;

A140: for v2,v0,v1 holds
  v0"\/"(v1"/\"(v2"/\"(v1"\/"v0))) = v0"\/"(v1"/\"v2)
proof let v2,v0,v1;
  v0"\/"((v1"/\"v2)"\/"(v1"/\"(v2"/\"v0))) = v0"\/"(v1"/\"v2) by A103;
  hence thesis by A138;
end;

A143: for v101,v2,v102 holds
 (v102"/\"v2)"\/"((v101"\/"(v102"/\"v2))"/\"v102) =
   (v102"/\"v2)"\/"(v102"/\"(v101"/\"(v102"\/"v2)))
proof let v101,v2,v102;
  v101"/\"(v102"\/"(v102"/\"v2)) =
    v102"/\"(v101"/\"(v102"\/"v2)) by A111;
  hence thesis by A57;
end;

A146: for v2,v1,v0 holds
  (v0"/\"v1)"\/"(v0"/\"(v2"\/"(v0"/\"v1))) =
     (v0"/\"v1)"\/"(v0"/\"(v2"/\"(v0"\/"v1)))
proof let v2,v1,v0;
  (v2"\/"(v0"/\"v1))"/\"v0 = v0"/\"(v2"\/"(v0"/\"v1)) by A4;
  hence thesis by A143;
end;

A148: for v2,v1,v0 holds
  v0"/\"(v2"\/"(v0"/\"v1)) =
    (v0"/\"v1)"\/"(v0"/\"(v2"/\"(v0"\/"v1)))
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v2"\/"(v0"/\"v1))) =
    v0"/\"(v2"\/"(v0"/\"v1)) by A119;
  hence thesis by A146;
end;

A154: for v102,v2,v101 holds
 (v101"/\"v2)"\/"(v101"/\"(v101"/\"(v102"/\"(v101"\/"v2)))) =
    (v101"/\"v2)"\/"(v101"/\"v102)
proof let v102,v2,v101;
  v102"/\"(v101"\/"(v101"/\"v2)) = v101"/\"(v102"/\"(v101"\/"v2)) by A111;
  hence thesis by A140;
end;

A157: for v2,v1,v0 holds
 (v0"/\"v1)"\/"(v0"/\"(v2"/\"(v0"\/"v1))) = (v0"/\"v1)"\/"(v0"/\"v2)
proof let v2,v1,v0;
  v0"/\"(v0"/\"(v2"/\"(v0"\/"v1))) = v0"/\"(v2"/\"(v0"\/"v1)) by A24;
  hence thesis by A154;
end;

A159: for v2,v1,v0 holds
   (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v2"\/"(v0"/\"v1))
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v0"/\"(v2"/\"(v0"\/"v1))) =
    (v0"/\"v1)"\/"(v0"/\"v2) by A157;
  hence thesis by A148;
end;

  let v1,v2,v3;
  v1"/\"(v2"\/"(v1"/\"v3)) = (v1"/\"v3)"\/"(v1"/\"v2) by A159;
  hence thesis by A8;
end;
