
theorem
  for F being non empty Graph-yielding Function, S being GraphSum of F
  st S is connected
  ex x being object, G being connected _Graph st F = x .--> G
proof
  let F be non empty Graph-yielding Function, S be GraphSum of F;
  assume A1: S is connected;
  consider G9 being GraphUnion of rng canGFDistinction F such that
    A2: S is G9-Disomorphic by Def27;
  consider M being PGraphMapping of G9, S such that
    A3: M is Disomorphism by A2, GLIB_010:def 24;
  G9 is connected by A1, A3, GLIB_010:140;
  then consider H being _Graph such that
    A4: rng canGFDistinction F = {H} by Th59;
  consider x being object such that
    A5: canGFDistinction F = x .--> H by A4, GLIBPRE1:4;
  A6: {x} = dom{[x,H]} by RELAT_1:9
    .= dom canGFDistinction F by A5, FUNCT_4:82
    .= dom F by Def25;
  then reconsider x0 = x as Element of dom F by TARSKI:def 1;
  A7: F = x .--> F.x0 by A6, DICKSON:1;
  then S is F.x0-Disomorphic by Th120;
  then consider M being PGraphMapping of F.x0,S such that
    A8: M is Disomorphism by GLIB_010:def 24;
  reconsider G = F.x0 as connected _Graph by A1, A8, GLIB_010:140;
  take x, G;
  thus thesis by A7;
end;
