reserve M,N for AbGroup;
 reserve R for Ring;
 reserve r for Element of R;
reserve M,N for LeftMod of R;
reserve f,g,h for Element of Funcs(the carrier of M, the carrier of N);
reserve a,b for Element of the carrier of R;
reserve R for comRing;
reserve M,M1,N,N1 for LeftMod of R;

theorem Th33: ::(a*b)*m by VECTSP_1:def 16 .= (b*a)*m
    for R be comRing, M be LeftMod of R, a be Element of R holds
    (curry (the lmult of M)).a is Endomorphism of R,M
    proof
      let R be comRing, M be LeftMod of R, a be Element of R;
      set f = (curry (the lmult of M)).a;
A1:   for m1,m2 being Element of M holds f.(m1+m2) = f.m1 + f.m2
      proof
        let m1,m2 be Element of M;
A2:     a*m1 = f.m1 & a*m2 = f.m2 by LOPBAN_8:7;
        f.(m1+m2) = a*(m1+m2) by LOPBAN_8:7
        .= f.m1 + f.m2 by A2,VECTSP_1:def 14;
      hence thesis;
    end;
    for b being Element of R, m being Element of M holds f.(b*m) = b*f.m
    proof
      let b be Element of R, m be Element of M;
      f.(b*m) = a*(b*m) by LOPBAN_8:7
        .= (a*b)*m by VECTSP_1:def 16 .= b*(a*m) by VECTSP_1:def 16
        .= b*f.m by LOPBAN_8:7;
      hence thesis;
    end; then
    f is homogeneous;
    hence thesis by A1,Def10,VECTSP_1:def 20;
  end;
