reserve G, G1, G2 for _Graph, H for Subgraph of G;

theorem
  G2 is G1-isomorphic implies G1.allSG(),G2.allSG() are_isomorphic
proof
  assume G2 is G1-isomorphic;
  then consider F being PGraphMapping of G1, G2 such that
    A1: F is isomorphism by GLIB_010:def 23;
  A2: dom SG2SGFunc F = G1.allSG() by FUNCT_2:def 1;
  A3: rng SG2SGFunc F = G2.allSG() by A1, Th32;
  A4: SG2SGFunc F is one-to-one by A1, Th31;
  now
    let G be _Graph;
    assume G in G1.allSG();
    then reconsider H = G as plain Subgraph of G1 by Th1;
    reconsider F9 = F|H as PGraphMapping of H, rng(F|H) by GLIBPRE1:88;
    A5: (SG2SGFunc F).G = rng(F | H) by Def5;
    F9 is isomorphism by A1, GLIBPRE1:110;
    hence (SG2SGFunc F).G is G-isomorphic _Graph
      by A5, GLIB_010:def 23;
  end;
  hence thesis by A2, A3, A4, GLIB_015:def 13;
end;
