reserve w, w1, w2 for Element of ExtREAL;
reserve c, c1, c2 for Complex;
reserve A, B, C, D for complex-membered set;
reserve F, G, H, I for ext-real-membered set;
reserve a, b, s, t, z for Complex;
reserve f, g, h, i, j for ExtReal;
reserve r for Real;
reserve e for set;

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;
