
theorem Th10:
  INT.Ring 1 is degenerated & INT.Ring 1 is Ring & INT.Ring 1 is
  almost_left_invertible unital distributive commutative
proof
  set n = 1, R = INT.Ring n;
A1: for x being Element of R st x <> 0.R ex y be Element of R st y*x = 1.R
  proof
    let x be Element of R;
    assume x <> 0.R;
    then x <> 0 by SUBSET_1:def 8;
    hence thesis by CARD_1:49,TARSKI:def 1;
  end;
A2: for a,b being Element of R holds a + b = b + a
  proof
    let a,b be Element of R;
    thus a + b = 0 by CARD_1:49,TARSKI:def 1
      .= b + a by CARD_1:49,TARSKI:def 1;
  end;
A3: for a being Element of R holds a + 0.R = a
  proof
    let a be Element of R;
    a = 0 by CARD_1:49,TARSKI:def 1;
    hence thesis by CARD_1:49,TARSKI:def 1;
  end;
A4: for a,b,c being Element of R holds (a * b) * c = a * (b * c)
  proof
    let a,b,c be Element of R;
    thus (a * b) * c = 0 by CARD_1:49,TARSKI:def 1
      .= a * (b * c) by CARD_1:49,TARSKI:def 1;
  end;
A5: for a being Element of R holds a + (-a) = 0.R
  proof
    let a be Element of R;
    thus a + (-a) = 0 by CARD_1:49,TARSKI:def 1
      .= 0.R by CARD_1:49,TARSKI:def 1;
  end;
A6: R is right_complementable
  proof
    let v be Element of R;
    take -v;
    thus thesis by A5;
  end;
A7: for a,b,c being Element of R holds (a + b) + c = a + (b + c)
  proof
    let a,b,c be Element of R;
    thus (a + b) + c = 0 by CARD_1:49,TARSKI:def 1
      .= a + (b + c) by CARD_1:49,TARSKI:def 1;
  end;
A8: for a being Element of R holds 1.R * a = a & a * 1.R = a
  proof
    let a be Element of R;
A9: a * 1.R = 0 by CARD_1:49,TARSKI:def 1
      .= a by CARD_1:49,TARSKI:def 1;
    1.R * a = 0 by CARD_1:49,TARSKI:def 1
      .= a by CARD_1:49,TARSKI:def 1;
    hence thesis by A9;
  end;
A10: R is well-unital
  by A8;
A11: for a,b being Element of R holds a * b = b * a
  proof
    let a,b be Element of R;
    thus a * b = 0 by CARD_1:49,TARSKI:def 1
      .= b * a by CARD_1:49,TARSKI:def 1;
  end;
A12: for a,b,c being Element of R holds a * (b + c) = a * b + a * c
  proof
    let a,b,c be Element of R;
    thus a * (b + c) = 0 by CARD_1:49,TARSKI:def 1
      .= a * b + a * c by CARD_1:49,TARSKI:def 1;
  end;
A13: for a,b,c being Element of R holds (b + c) * a = b * a + c * a
  proof
    let a,b,c be Element of R;
    thus (b + c) * a = 0 by CARD_1:49,TARSKI:def 1
      .= b * a + c * a by CARD_1:49,TARSKI:def 1;
  end;
  0.R = 0 by CARD_1:49,TARSKI:def 1
    .= 1.R by CARD_1:49,TARSKI:def 1;
  hence thesis by A1,A2,A11,A7,A4,A3,A13,A12,A6,A10,GROUP_1:def 3,RLVECT_1:def
2,def 3,def 4,VECTSP_1:def 7;
end;
