theorem Th17:
  - (g + h) = (-g) - h
proof
A1: dom -(g+h) = dom(g+h) by VALUED_1:8;
  dom(g+h) = dom g /\ dom h & dom(-g-h) = dom(-g) /\ dom h by VALUED_1:12,def 1
;
  hence
A2: dom -(g+h) = dom(-g-h) by A1,VALUED_1:8;
  let x be object;
  assume
A3: x in dom -(g+h);
  thus (-(g+h)).x = -(g+h).x by VALUED_1:8
    .= -(g.x+h.x) by A1,A3,VALUED_1:def 1
    .= -g.x-h.x
    .= (-g).x-h.x by VALUED_1:8
    .= (-g-h).x by A2,A3,VALUED_1:13;
end;
