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

theorem ThQLT2:
(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 holds v0"\/"v0 = v0) &
(for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
(for v0,v2,v1 holds (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
(for v0,v2,v1 holds v0"/\"(v1"\/"v2) = (v0"/\"v1)"\/"(v0"/\"v2)) implies
  for v1,v2,v3 holds v1"\/"(v2"/\"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 holds v0"\/"v0 = v0;
assume A6: for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2);
assume A7: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2);

A9: 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 A7;
end;

assume A10: for v0,v2,v1 holds
  v0"/\"(v1"\/"v2) = (v0"/\"v1)"\/"(v0"/\"v2);

A15: for v102,v101 holds v101"/\"v102 = v101"/\"(v101"/\"v102)
proof let v102,v101;
  v101"/\"v101 = v101 by A2;
  hence thesis by A3;
end;

A20: for v102,v101 holds
  v101"\/"v102 = v101"\/"(v101"\/"v102)
proof let v102,v101;
  v101"\/"v101 = v101 by A5;
  hence thesis by A6;
end;

A24: for v2,v0,v1 holds
  (v1"/\"v0)"\/"(v0"/\"v2) = v0"/\"(v1"\/"v2)
proof let v2,v0,v1;
  v0"/\"v1 = v1"/\"v0 by A4;
  hence thesis by A10;
end;

A28: for v102,v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"v102 = (v0"/\"v1)"\/"((v0"/\"v2)"\/"v102)
proof let v102,v0,v2,v1;
  (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"v2) by A10;
  hence thesis by A6;
end;

A32: for v102,v1,v100 holds
 (v100"/\"v1)"\/"(v100"/\"v102) = v100"/\"((v100"/\"v1)"\/"v102)
proof let v102,v1,v100;
  v100"/\"(v100"/\"v1) = v100"/\"v1 by A15;
  hence thesis by A10;
end;

A35: for v0,v2,v1 holds v0"/\"(v1"\/"v2) = v0"/\"((v0"/\"v1)"\/"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"v2) by A10;
  hence thesis by A32;
end;

A39: for v1,v101,v100 holds
 (v100"/\"v101)"\/"(v100"/\"v1) = v100"/\"(v101"\/"(v100"/\"v1))
proof let v1,v101,v100;
  v100"/\"(v100"/\"v1) = v100"/\"v1 by A15;
  hence thesis by A10;
end;

A42: for v0,v2,v1 holds v0"/\"(v1"\/"v2) = v0"/\"(v1"\/"(v0"/\"v2))
proof let v0,v2,v1;
  (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"v2) by A10;
  hence thesis by A39;
end;

A46: for v102,v1,v100 holds
 (v100"\/"v1)"/\"(v100"\/"((v100"\/"v1)"/\"v102)) =
    v100"\/"((v100"\/"v1)"/\"v102)
proof let v102,v1,v100;
  v100"\/"(v100"\/"v1) = v100"\/"v1 by A20;
  hence thesis by A9;
end;

A49: for v2,v1,v0 holds
  (v0"\/"v1)"/\"(v0"\/"v2) = v0"\/"((v0"\/"v1)"/\"v2)
proof let v2,v1,v0;
  (v0"\/"v1)"/\"(v0"\/"((v0"\/"v1)"/\"v2)) =
    (v0"\/"v1)"/\"(v0"\/"v2) by A42;
  hence thesis by A46;
end;

A53: for v2,v1,v101 holds
 (v101"\/"v1)"/\"(((v101"\/"v1)"/\"v101)"\/"(v1"/\"v2)) =
    v101"\/"(v1"/\"v2)
proof let v2,v1,v101;
  (v101"\/"v1)"/\"(v101"\/"(v1"/\"v2)) = v101"\/"(v1"/\"v2) by A9;
  hence thesis by A35;
end;

A56: for v2,v1,v0 holds
  (v0"\/"v1)"/\"((v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"\/"v1)"/\"v0 = v0"/\"(v0"\/"v1) by A4;
  hence thesis by A53;
end;

A58: for v2,v1,v0 holds
  (v0"\/"v1)"/\"((v0"/\"v0)"\/"((v0"/\"v1)"\/"(v1"/\"v2))) =
    v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"/\"(v0"\/"v1))"\/"(v1"/\"v2) =
    (v0"/\"v0)"\/"((v0"/\"v1)"\/"(v1"/\"v2)) by A28;
  hence thesis by A56;
end;

A60: for v2,v1,v0 holds
  (v0"\/"v1)"/\"(v0"\/"((v0"/\"v1)"\/"(v1"/\"v2))) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  v0"/\"v0 = v0 by A2;
  hence thesis by A58;
end;

A62: for v1,v2,v0 holds
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"(v0"\/"v2))) = v0"\/"(v1"/\"v2)
proof let v1,v2,v0;
  (v0"/\"v1)"\/"(v1"/\"v2) = v1"/\"(v0"\/"v2) by A24;
  hence thesis by A60;
end;

A64: for v1,v2,v0 holds
  v0"\/"(v1"/\"(v0"\/"v2)) = v0"\/"(v1"/\"v2)
proof let v1,v2,v0;
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"(v0"\/"v2))) =
    v0"\/"(v1"/\"(v0"\/"v2)) by A9;
  hence thesis by A62;
end;

A66: for v1,v2,v0 holds
  v0"\/"((v0"\/"v2)"/\"v1) = v0"\/"(v1"/\"v2)
proof let v1,v2,v0;
  v1"/\"(v0"\/"v2) = (v0"\/"v2)"/\"v1 by A4;
  hence thesis by A64;
end;

A69: for v2,v1,v0 holds
  (v0"\/"v1)"/\"(v0"\/"v2) = v0"\/"(v2"/\"v1)
proof let v2,v1,v0;
  v0"\/"((v0"\/"v1)"/\"v2) = (v0"\/"v1)"/\"(v0"\/"v2) by A49;
  hence thesis by A66;
end;

  let v1,v2,v3;
  v1"\/"(v2"/\"v3) = (v1"\/"v3)"/\"(v1"\/"v2) by A69;
  hence thesis by A4;
end;
