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
  f <> {} & f is non-empty & f [-] c1 = f [-] c2 implies c1 = c2
proof
  assume that
A1: f <> {} and
A2: f is non-empty and
A3: f[-]c1 = f[-]c2;
  consider x being object such that
A4: x in dom f by A1,XBOOLE_0:def 1;
  f.x in rng f by A4,FUNCT_1:def 3;
  then
A5: f.x <> {} by A2,RELAT_1:def 9;
  dom f = dom(f[-]c2) by Def37;
  then
A6: (f[-]c2).x = f.x-c2 by A4,Def37;
  dom f = dom(f[-]c1) by Def37;
  then (f[-]c1).x = f.x-c1 by A4,Def37;
  hence c1 = c2 by A3,A5,A6,Th8;
end;
