theorem Th196:
  a <> 0 & a**A c= a**B implies A c= B
proof
  assume that
A1: a <> 0 and
A2: a**A c= a**B;
  let z;
  assume z in A;
  then a*z in a**A by Th193;
  then ex c st a*z = a*c & c in B by A2,Th195;
  hence thesis by A1,XCMPLX_1:5;
end;
