reserve x, y for object, X, X1, X2 for set;
reserve Y, Y1, Y2 for complex-functions-membered set,
  c, c1, c2 for Complex,
  f for PartFunc of X,Y,
  f1 for PartFunc of X1,Y1,
  f2 for PartFunc of X2, Y2,
  g, h, k for complex-valued Function;

theorem Th33:
  g <> {} & g is non-empty & g (/) c1 = g (/) c2 implies c1 = c2
proof
  assume that
A1: g <> {} and
A2: g is non-empty and
A3: g(/)c1 = g(/)c2;
  consider x being object such that
A4: x in dom g by A1,XBOOLE_0:def 1;
  g.x in rng g by A4,FUNCT_1:def 3;
  then
A5: g.x <> {} by A2,RELAT_1:def 9;
  (g(/)c1).x = g.x/c1 & (g(/)c2).x = g.x/c2 by VALUED_1:6;
  then c1" = c2" by A3,A5,XCMPLX_1:5;
  hence c1 = c2 by XCMPLX_1:201;
end;
