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

theorem Th13:
    for M,N be LeftMod of R, f be Homomorphism of R,M,N holds
    f is one-to-one iff ker(f) = {0.M}
    proof
      let M,N be LeftMod of R, f be Homomorphism of R,M,N;
A1:   f.0.M = f.(0.M+0.M) by RLVECT_1:4 .= f.0.M + f.0.M
      by Def10,VECTSP_1:def 20; then
A2:   f.0.M = 0.N by RLVECT_1:9;
A3:   0.M in ker f by A2;
      thus f is one-to-one implies ker(f) = {0.M}
      proof
        assume
A5:     f is one-to-one;
        for x being object holds x in ker f iff x in {0.M}
        proof
          let x be object;
          x in ker f implies x in {0.M}
          proof
            assume x in ker f; then
            consider x1 be Element of M such that
A7:         x1 = x & f.x1 = 0.N;
A8:         f.x1 = f.0.M by A1,A7,RLVECT_1:9;
            dom f = the carrier of M by FUNCT_2:def 1; then
            x = 0.M by A5,A8,A7;
            hence thesis by TARSKI:def 1;
          end;
          hence thesis by A3,TARSKI:def 1;
        end;
        hence thesis by TARSKI:2;
      end;
        assume
A10:    ker(f) = {0.M};
        for x1,x2 be object st x1 in dom f & x2 in dom f & f.x1 = f.x2
        holds x1 = x2
        proof
          let x1,x2 be object;
          assume
A11:      x1 in dom f & x2 in dom f & f.x1 = f.x2; then
          reconsider x1,x2 as Element of M;
A12:      -(f.x2) = (-1.R)*(f.x2) by VECTSP_1:14
          .=f.((-1.R)*x2) by Def10,MOD_2:def 2 .= f.(-x2) by VECTSP_1:14;
          0.N = f.x1 + (f.-x2) by A12,A11,RLVECT_1:5
          .= f.(x1+(-x2)) by Def10,VECTSP_1:def 20; then
          x1 - x2 in ker f; then
          x1 - x2 = 0.M by A10,TARSKI:def 1;
          hence thesis by VECTSP_1:19;
        end;
        hence thesis;
    end;
