reserve M,N for AbGroup;
 reserve R for Ring;
 reserve r for Element of R;

theorem Th14:
    for R for M be AbGroup, s be Function of R,End_Ring(M) st
    s is RingHomomorphism holds AbGrLMod(M,s) is LeftMod of R
    proof
      let R; let M be AbGroup,
      s be Function of R,End_Ring(M);
      assume
A1:   s is RingHomomorphism;
A2:   AbGrLMod(M,s) is Abelian
      proof
        for x,y be Element of AbGrLMod(M,s) holds x+y = y+x
        proof
          let x, y be Element of AbGrLMod(M,s);
        reconsider x1 = x, y1 = y as Element of M;
          x + y = x1 + y1 .= y1 + x1 .= y + x;
          hence thesis;
        end;
        hence thesis;
      end;
A3:   AbGrLMod(M,s) is add-associative
      proof
        for x,y,z be Element of AbGrLMod(M,s) holds (x+y) +z = x +(y+z)
        proof
          let x, y, z be Element of AbGrLMod(M,s);
          reconsider x1 = x, y1 = y, z1 = z as Element of M;
          reconsider xy = x+y as Element of AbGrLMod(M,s);
          reconsider xy1 = xy, yz1 = y + z as Element of M;
          reconsider yh = y1 + y1 as Element of M;
          (x+y)+z = (x1+y1)+z1 .= x1+(y1+z1) by ALGSTR_1:7
            .= x + (y+z);
          hence thesis;
        end;
        hence thesis;
      end;
A4:   AbGrLMod(M,s) is right_zeroed
      proof
        for x be Element of AbGrLMod(M,s) holds x + 0.AbGrLMod(M,s) = x
        proof
          let x be Element of AbGrLMod(M,s);
        reconsider x1 = x as Element of M;
          x + 0.AbGrLMod(M,s) = x1 + 0.M .= x by ALGSTR_1:7;
          hence thesis;
        end;
        hence thesis;
      end;
A5:   AbGrLMod(M,s) is right_complementable
      proof
        for x being Element of AbGrLMod(M,s) holds x is right_complementable
        proof
          let x be Element of AbGrLMod(M,s);
          reconsider x1 = x as Element of M;
          consider y1 be Element of M such that
A6:       x1 + y1 = 0.M by ALGSTR_1:7;
          reconsider y = y1 as Element of AbGrLMod(M,s);
          take y;
          thus thesis by A6;
        end;
        hence thesis;
      end;
      AbGrLMod(M,s) is scalar-unital
      proof
        for x be Element of AbGrLMod(M,s) holds 1.R * x = x
        proof
          let x be Element of AbGrLMod(M,s);
          reconsider x1 = x as Element of M;
          consider h be additive Function of the carrier of M,the carrier of M
          such that
A7:       h = s.(1.R) & 1.R * x = h.x by Def12;
          s is unity-preserving by A1;
          hence thesis by A7;
        end;
        hence thesis;
      end; then
      AbGrLMod(M,s) is Abelian add-associative right_zeroed
      right_complementable vector-distributive scalar-distributive
      scalar-associative scalar-unital by A2,A3,A4,A5,A1,Lm12,Lm13,Lm14;
      hence thesis;
    end;
