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

theorem ThQLT4:
(for v0 holds v0"/\"v0 = v0) &
(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 v0,v2,v1 holds
   (v0"\/"(v1"/\"v2))"\/"((v0"\/"v1)"/\"v2) = v0"\/"(v1"/\"v2)) implies
for v1,v2,v3 holds v1"\/"(v2"/\"v3) = (v1"\/"v2)"/\"(v1"\/"v3)
proof
assume A1: for v0 holds v0"/\"v0 = v0;
assume A2: for v1,v0 holds v0"/\"v1 = v1"/\"v0;
assume A3: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2);
assume A4: for v0 holds v0"\/"v0 = v0;
assume A5: for v2,v1,v0 holds
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2);
assume A6: for v1,v0 holds v0"\/"v1 = v1"\/"v0;
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 A2;
  hence thesis by A7;
end;

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

A12: for v2,v1,v0 holds
  v0"\/"((v1"/\"v2)"\/"((v0"\/"v1)"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"\/"(v1"/\"v2))"\/"((v0"\/"v1)"/\"v2) =
    v0"\/"((v1"/\"v2)"\/"((v0"\/"v1)"/\"v2)) by A5;
hence thesis by A10;
end;

A16: 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 A6;
  hence thesis by A3;
end;

A19: for v102,v100 holds (v100"\/"v102)"\/"v102 = v100"\/"v102
proof let v102,v100;
  v102"\/"v102 = v102 by A4;
  hence thesis by A5;
end;

A22: for v1,v0 holds v1"\/"(v0"\/"v1) = v0"\/"v1
proof let v1,v0;
  (v0"\/"v1)"\/"v1 = v1"\/"(v0"\/"v1) by A6;
  hence thesis by A19;
end;

A25: for v2,v0,v1 holds (v1"\/"v0)"\/"v2 = v0"\/"(v1"\/"v2)
proof let v2,v0,v1;
  v0"\/"v1 = v1"\/"v0 by A6;
  hence thesis by A5;
end;

A28: for v0,v2,v1 holds
  v0"\/"(v1"\/"v2) = v1"\/"(v0"\/"v2)
proof let v0,v2,v1;
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2) by A5;
  hence thesis by A25;
end;

A31: for v102,v101 holds
  v101"/\"(v101"\/"(v101"/\"v102)) = v101"\/"(v101"/\"v102)
proof let v102,v101;
  v101"\/"v101 = v101 by A4;
  hence thesis by A9;
end;

A34: for v2,v1,v0 holds
  v0"\/"((v1"/\"v2)"\/"(v2"/\"(v0"\/"v1))) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"\/"v1)"/\"v2 = v2"/\"(v0"\/"v1) by A2;
  hence thesis by A12;
end;

A37: for v102,v101 holds
  v101"\/"(v101"/\"(v101"\/"v102)) = v101"/\"(v101"\/"v102)
proof let v102,v101;
  v101"/\"v101 = v101 by A1;
  hence thesis by A16;
end;

A40: for v0,v2,v1 holds
  (v1"/\"v0)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  v0"/\"v1 = v1"/\"v0 by A2;
  hence thesis by A16;
end;

A44: for v101,v2,v1 holds
 ((v101"\/"v1)"/\"v101)"\/"(v101"\/"(v1"/\"v2)) =
    (v101"\/"v1)"/\"(v101"\/"(v1"/\"v2))
proof let v101,v2,v1;
  (v101"\/"v1)"/\"(v101"\/"(v1"/\"v2)) = v101"\/"(v1"/\"v2) by A9;
  hence thesis by A16;
end;

A47: for v0,v2,v1 holds
  (v0"/\"(v0"\/"v1))"\/"(v0"\/"(v1"/\"v2)) =
    (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2))
proof let v0,v2,v1;
  (v0"\/"v1)"/\"v0 = v0"/\"(v0"\/"v1) by A2;
  hence thesis by A44;
end;

A49: for v2,v1,v0 holds
 v0"\/"((v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)) = (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2))
proof let v2,v1,v0;
 (v0"/\"(v0"\/"v1))"\/"(v0"\/"(v1"/\"v2)) =
    v0"\/"((v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)) by A28;
hence thesis by A47;
end;

A51: for v2,v1,v0 holds
  v0"\/"((v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2) by A9;
  hence thesis by A49;
end;

A53: for v0,v1 holds
  v0"/\"(v0"\/"(v1"/\"v0)) = v0"\/"(v0"/\"v1)
proof let v0,v1;
  v0"/\"v1 = v1"/\"v0 by A2;
  hence thesis by A31;
end;

A56: for v102,v1,v100 holds
 (v100"/\"(v100"\/"v1))"\/"v102 = v100"\/"((v100"/\"(v100"\/"v1))"\/"v102)
proof let v102,v1,v100;
  v100"\/"(v100"/\"(v100"\/"v1)) = v100"/\"(v100"\/"v1) by A37;
  hence thesis by A5;
end;

A60: for v2,v1,v0 holds (v0"/\"(v0"\/"v1))"\/"(v1"/\"v2) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  v0"\/"((v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)) = (v0"/\"(v0"\/"v1))"\/"(v1"/\"v2)
    by A56;
  hence thesis by A51;
end;

A63: for v101,v100,v1 holds
  (v100"/\"v101)"\/"(v101"/\"(v1"\/"v100)) = v101"/\"(v100"\/"(v1"\/"v100))
proof let v101,v100,v1;
  v100"\/"(v1"\/"v100) = v1"\/"v100 by A22;
  hence thesis by A40;
end;

A66: for v1,v0,v2 holds
  (v0"/\"v1)"\/"(v1"/\"(v2"\/"v0)) = v1"/\"(v2"\/"v0)
proof let v1,v0,v2;
  v0"\/"(v2"\/"v0) = v2"\/"v0 by A22;
  hence thesis by A63;
end;

A68: for v2,v1,v0 holds
  v0"\/"(v2"/\"(v0"\/"v1)) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v1"/\"v2)"\/"(v2"/\"(v0"\/"v1)) = v2"/\"(v0"\/"v1) by A66;
  hence thesis by A34;
end;

A71: for v1,v2,v0 holds v0"\/"((v0"\/"v2)"/\"v1) = v0"\/"(v2"/\"v1)
proof let v1,v2,v0;
  v1"/\"(v0"\/"v2) = (v0"\/"v2)"/\"v1 by A2;
  hence thesis by A68;
end;

A74: for v1,v2,v0 holds
  v0"\/"(v1"/\"(v0"\/"v2)) = v0"\/"(v1"/\"v2)
proof let v1,v2,v0;
  v2"/\"v1 = v1"/\"v2 by A2;
  hence thesis by A68;
end;

A77: for v100,v2 holds
 v100"/\"(v100"\/"(v2"/\"v100)) = v100"\/"(v100"/\"(v100"\/"v2))
proof let v100,v2;
  v100"\/"(v100"/\"(v100"\/"v2)) = v100"\/"(v2"/\"v100) by A68;
  hence thesis by A31;
end;

A80: for v1,v0 holds v0"\/"(v0"/\"v1) = v0"\/"(v0"/\"(v0"\/"v1))
proof let v1,v0;
  v0"/\"(v0"\/"(v1"/\"v0)) = v0"\/"(v0"/\"v1) by A53;
  hence thesis by A77;
end;

A82: for v1,v0 holds v0"\/"(v0"/\"v1) = v0"/\"(v0"\/"v1)
proof let v1,v0;
  v0"\/"(v0"/\"(v0"\/"v1)) = v0"/\"(v0"\/"v1) by A37;
  hence thesis by A80;
end;

A85: for v2,v1,v0 holds
  (v0"\/"(v0"/\"v1))"\/"(v1"/\"v2) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  v0"/\"(v0"\/"v1) = v0"\/"(v0"/\"v1) by A82;
  hence thesis by A60;
end;

A87: for v2,v1,v0 holds
  v0"\/"((v0"/\"v1)"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2)
proof let v2,v1,v0;
  (v0"\/"(v0"/\"v1))"\/"(v1"/\"v2) = v0"\/"((v0"/\"v1)"\/"(v1"/\"v2)) by A5;
  hence thesis by A85;
end;

A90: for v102,v1,v100 holds
 (v100"\/"(v100"/\"v1))"\/"((v100"\/"v1)"/\"(v100"\/"v102)) =
   (v100"\/"v1)"/\"(v100"\/"v102)
proof let v102,v1,v100;
  v100"/\"(v100"\/"v1) = v100"\/"(v100"/\"v1) by A82;
  hence thesis by A40;
end;

A93: for v2,v1,v0 holds
 v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"(v0"\/"v2))) =
   (v0"\/"v1)"/\"(v0"\/"v2)
proof let v2,v1,v0;
  (v0"\/"(v0"/\"v1))"\/"((v0"\/"v1)"/\"(v0"\/"v2)) =
     v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"(v0"\/"v2))) by A5;
  hence thesis by A90;
end;

A96: for v100,v101,v2,v1 holds
 v100"\/"(v101"\/"(v1"/\"v2)) = v101"\/"(v100"\/"((v101"\/"v1)"/\"v2))
proof let v100,v101,v2,v1;
  v101"\/"((v101"\/"v1)"/\"v2) = v101"\/"(v1"/\"v2) by A71;
  hence thesis by A28;
end;

A101: for v1,v2,v0 holds
 (v0"/\"v1)"\/"(v0"\/"(v1"/\"(v0"\/"v2))) = (v0"\/"v1)"/\"(v0"\/"v2)
proof let v1,v2,v0;
  v0"\/"((v0"/\"v1)"\/"((v0"\/"v1)"/\"(v0"\/"v2))) =
    (v0"/\"v1)"\/"(v0"\/"(v1"/\"(v0"\/"v2))) by A96;
  hence thesis by A93;
end;

A103: for v0,v2,v1 holds
 (v0"/\"v1)"\/"(v0"\/"(v1"/\"v2)) = (v0"\/"v1)"/\"(v0"\/"v2)
proof let v0,v2,v1;
  v0"\/"(v1"/\"(v0"\/"v2)) = v0"\/"(v1"/\"v2) by A74;
  hence thesis by A101;
end;

A105: for v2,v1,v0 holds
 v0"\/"((v0"/\"v1)"\/"(v1"/\"v2)) = (v0"\/"v1)"/\"(v0"\/"v2)
proof let v2,v1,v0;
  (v0"/\"v1)"\/"(v0"\/"(v1"/\"v2)) = v0"\/"((v0"/\"v1)"\/"(v1"/\"v2)) by A28;
  hence thesis by A103;
end;

for v0,v2,v1 holds v0"\/"(v1"/\"v2) = (v0"\/"v1)"/\"(v0"\/"v2)
proof let v0,v2,v1;
  v0"\/"((v0"/\"v1)"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2) by A87;
  hence thesis by A105;
end;
hence thesis;
end;
