
theorem Th4:
  for L being right_unital associative non trivial doubleLoopStr
  for a,b being Element of L st b is left_invertible right_mult-cancelable
  & b*(/b) = (/b)*b holds a*b/b = a
  proof
    let L be right_unital associative non trivial doubleLoopStr;
    let a,b be Element of L;
    assume
A1: b is left_invertible right_mult-cancelable;
    assume b*(/b) = (/b)*b;
    then b*(/b) = 1.L by A1,ALGSTR_0:def 30;
    hence a = a*(b*(/b))
    .= a*b/b by GROUP_1:def 3;
  end;
