
theorem Th120:
  for x being object, G being _Graph, S being GraphSum of x .--> G
  holds S is G-Disomorphic
proof
  let x be object, G be _Graph, S be GraphSum of x .--> G;
  x in {x} by TARSKI:def 1;
  then x in dom{[x,G]} by RELAT_1:9;
  then reconsider x0 = x as Element of dom(x .--> G) by FUNCT_4:82;
  set H = replaceVerticesEdges(
    renameElementsDistinctlyFunc(the_Vertices_of (x .--> G),x0),
    renameElementsDistinctlyFunc(the_Edges_of (x .--> G),x0));
  A1: (canGFDistinction(x .--> G)).x = H by Def25;
  dom canGFDistinction(x .--> G) = dom (x .--> G) by Def25
    .= dom{[x,G]} by FUNCT_4:82
    .= {x} by RELAT_1:9;
  then canGFDistinction(x .--> G) = x .--> H by A1, DICKSON:1;
  then A2: rng canGFDistinction(x .--> G) = {H} by FUNCOP_1:88;
  consider G9 being GraphUnion of rng canGFDistinction(x .--> G) such that
    A3: S is G9-Disomorphic by Def27;
  H is ((x .--> G).x0)-Disomorphic & (x .--> G).x = G by Th17, FUNCOP_1:72;
  then A4: H is G-Disomorphic;
  G9 == H by A2, GLIB_014:24;
  then G9 is H-Disomorphic by GLIBPRE0:76;
  then G9 is G-Disomorphic by A4;
  hence S is G-Disomorphic by A3;
end;
