reserve X,Y for set;
reserve R for domRing-like commutative Ring;
reserve c for Element of R;

theorem Th1:
  for R being domRing-like commutative Ring
  for a,b,c being Element of R holds a <> 0.R implies
    (a * b = a * c implies b = c) &
    (b * a = c * a implies b = c)
proof
  let R be domRing-like commutative Ring;
  let a,b,c be Element of R;
  assume
A1: a <> 0.R;
  now
    assume a * b = a * c;
    then 0.R = (a * b) + (-(a * c)) by RLVECT_1:def 10
      .= (a * b) + (a *(-c)) by VECTSP_1:8
      .= a * (b + (-c)) by VECTSP_1:def 2
      .= a * (b - c) by RLVECT_1:def 11;
    then b - c = 0.R by A1,VECTSP_2:def 1;
    then c = (b - c) + c by RLVECT_1:4
      .= (b + (-c)) + c by RLVECT_1:def 11
      .= b + (c + (-c)) by RLVECT_1:def 3
      .= b + 0.R by RLVECT_1:def 10
      .= b by RLVECT_1:4;
    hence b = c;
  end;
  hence thesis;
end;
