
theorem
  for F being associative commutative well-unital distributive
          almost_left_invertible non empty doubleLoopStr,
      x,y,z being Element of F holds
     (x <> 0.F & x*y = x*z) implies y = z
proof
  let F be associative commutative well-unital distributive
  almost_left_invertible non empty doubleLoopStr, x,y,z be Element of F;
  assume x<>0.F;
  then consider x1 being Element of F such that
A1: x1*x = 1.F by Def9;
A2: x1*x*y = x1*(x*y) & x1*(x*z) = x1*x*z by GROUP_1:def 3;
  assume x*y = x*z;
  then x*x1*y = z by A1,A2,Def8;
  hence thesis by A1;
end;
