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 Th8:
  g <> {} & g - c1 = g - c2 implies c1 = c2
proof
  assume that
A1: g <> {} and
A2: g-c1 = g-c2;
  consider x being object such that
A3: x in dom g by A1,XBOOLE_0:def 1;
  dom g = dom(g-c2) by VALUED_1:def 2;
  then
A4: (g-c2).x = g.x-c2 by A3,VALUED_1:def 2;
  dom g = dom(g-c1) by VALUED_1:def 2;
  then (g-c1).x = g.x-c1 by A3,VALUED_1:def 2;
  hence c1 = c2 by A2,A4;
end;
