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 Th9:
  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;
  hence c1 = c2 by A3,A5,XCMPLX_1:5;
end;
