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 Th144:
  A c= B iff a++A c= a++B
proof
  thus A c= B implies a++A c= a++B by Th47;
  assume
A1: a++A c= a++B;
  let z;
  assume z in A;
  then a+z in a++A by Th141;
  then ex c st a+z = a+c & c in B by A1,Th143;
  hence thesis;
end;
