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

theorem Th75:
  the_Vertices_of G.allSpanningSG() = { the_Vertices_of G }
proof
  now
    let x be object;
    hereby
      assume x in the_Vertices_of G.allSpanningSG();
      then consider H being _Graph such that
        A1: H in G.allSpanningSG() & x = the_Vertices_of H
        by GLIB_014:def 14;
      H is spanning Subgraph of G by A1, Th60;
      hence x = the_Vertices_of G by A1, GLIB_000:def 33;
    end;
    assume A2: x = the_Vertices_of G;
    set H = the plain spanning Subgraph of G;
    H in G.allSpanningSG() & the_Vertices_of H = the_Vertices_of G
      by Th60, GLIB_000:def 33;
    hence x in the_Vertices_of G.allSpanningSG() by A2, GLIB_014:def 14;
  end;
  hence thesis by TARSKI:def 1;
end;
