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

theorem
  the_Edges_of G.allSpanningSG() = bool the_Edges_of G
proof
  the_Edges_of G.allSpanningSG() c= the_Edges_of G.allSG()
    by GLIBPRE1:115;
  then A1: the_Edges_of G.allSpanningSG() c= bool the_Edges_of G
    by Th38;
  now
    let x be object;
    reconsider X = x as set by TARSKI:1;
    assume x in bool the_Edges_of G;
    then X c= the_Edges_of G;
    then A2: X c= G.edgesBetween(the_Vertices_of G) by GLIB_000:34;
    the_Vertices_of G c= the_Vertices_of G;
    then A3: the_Vertices_of G is non empty Subset of the_Vertices_of G;
    set H = the plain inducedSubgraph of G, the_Vertices_of G, X;
    the_Vertices_of H = the_Vertices_of G by A2, A3, GLIB_000:def 37;
    then H is spanning by GLIB_000:def 33;
    then A4: H in G.allSpanningSG() by Th60;
    the_Edges_of H = X by A2, A3, GLIB_000:def 37;
    hence x in the_Edges_of G.allSpanningSG() by A4, GLIB_014:def 15;
  end;
  then bool the_Edges_of G c= the_Edges_of G.allSpanningSG()
    by TARSKI:def 3;
  hence thesis by A1, XBOOLE_0:def 10;
end;
