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

theorem
  for F being almost_left_invertible commutative Ring
  for a, b being Element of F st
    a <> 0.F & b <> 0.F holds a"*b" = (b*a)"
proof
  let F be almost_left_invertible commutative Ring;
  let a,b be Element of F such that
A1: a <> 0.F and
A2: b <> 0.F;
A3: b*a * (a"*b") = b*a*a"*b" by GROUP_1:def 3
    .= b*(a*a")*b" by GROUP_1:def 3
    .= b*1_F*b" by A1,VECTSP_1:def 10
    .= b*b"
    .= 1_F by A2,VECTSP_1:def 10;
  b*a <> 0.F by A1,A2,VECTSP_1:12;
  hence thesis by A3,VECTSP_1:def 10;
end;
