
theorem Th11:
  for n be non zero Nat holds
  the addLoopStr of INT.Ring n is non empty Abelian right_complementable
  add-associative right_zeroed
proof
  let n be non zero Nat;
A1: 1 <= n by NAT_1:14;
  now per cases;
    suppose n=1;
      hence INT.Ring n is Ring by INT_3:10;
    end;
    suppose n<>1;
      then 1 < n by A1,XXREAL_0:1;
      hence INT.Ring n is Ring by INT_3:11;
    end;
  end;
  then reconsider R = INT.Ring n as Ring;
  set S = the addLoopStr of INT.Ring n;
A2: for v,w being Element of S holds v + w = w + v
  proof
    let v,w be Element of S;
    reconsider v1=v,w1=w as Element of R;
    thus v+w = v1+w1
      .= w1+v1
      .= w+v;
  end;
A3: for x being Element of S holds x is right_complementable
  proof
    let v be Element of S;
    reconsider v1=v as Element of R;
    consider w1 being Element of R such that
A4: v1 + w1 = 0.R by ALGSTR_0:def 11;
    reconsider w=w1 as Element of S;
    v+w = 0.S by A4;
    hence thesis;
  end;
A5: for u,v,w being Element of S holds (u + v) + w = u + (v + w)
  proof
    let u,v,w be Element of S;
    reconsider u1=u,v1=v,w1=w as Element of R;
    thus (u + v) + w = (u1 + v1) + w1
      .= u1 + (v1 + w1) by RLVECT_1:def 3
      .= u + (v + w);
  end;
A6: for v being Element of S holds v + (0.S) = v
  proof
    let v be Element of S;
    reconsider v1=v as Element of R;
    thus v + (0.S) = v1 + 0.R
      .= v;
  end;
  thus thesis by A2,A3,A5,A6,RLVECT_1:def 2,RLVECT_1:def 3,
  RLVECT_1:def 4;
end;
