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 Th10:
  - (g + c) = -g - c
proof
A1: dom -(g+c) = dom(g+c) by VALUED_1:8;
A2: dom(g+c) = dom g & dom(-g-c) = dom -g by VALUED_1:def 2;
  hence dom -(g+c) = dom(-g-c) by A1,VALUED_1:8;
  let x be object;
  assume
A3: x in dom -(g+c);
A4: dom -g = dom g by VALUED_1:8;
  thus (-(g+c)).x = -(g+c).x by VALUED_1:8
    .= -(g.x+c) by A1,A3,VALUED_1:def 2
    .= -g.x-c
    .= (-g).x-c by VALUED_1:8
    .= (-g-c).x by A2,A1,A4,A3,VALUED_1:def 2;
end;
