reserve R,R1 for commutative Ring;
reserve A,B for non degenerated commutative Ring;
reserve o,o1,o2 for object;
reserve r,r1,r2 for Element of R;
reserve a,a1,a2,b,b1 for Element of A;
reserve f for Function of R, R1;
reserve p for Element of Spectrum A;

theorem Th11:
  r1 in Unit_Set(R) implies r1 is right_mult-cancelable
  proof
    assume r1 in Unit_Set(R); then
    consider r2 such that
A2: r2*r1 = 1.R by Th10;
    for u,v being Element of R st u*r1 = v*r1 holds u = v
    proof
      let u,v be Element of R;
      assume u*r1 = v*r1; then
A5:   r1*(u - v) = r1*v - r1*v by VECTSP_1:11
      .= 0.R by RLVECT_1:15;
      u - v = (r2*r1)*(u - v) by A2
      .= r2*(r1*(u - v)) by GROUP_1:def 3 .= 0.R by A5;
      hence thesis by VECTSP_1:19;
    end;
    hence thesis;
  end;
