reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;
reserve V for set;

theorem Th92:
  for G2, V for G1 being addVertices of G2, V, v being Vertex of G1
  st v in V \ the_Vertices_of G2 holds v is isolated non cut-vertex
proof
  let G2, V;
  let G1 be addVertices of G2, V;
  let v be Vertex of G1;
  assume A1: v in V \ the_Vertices_of G2;
  v.edgesInOut() = {}
  proof
    assume v.edgesInOut() <> {};
    then consider e being object such that
      A2: e in v.edgesInOut() by XBOOLE_0:def 1;
    e in the_Edges_of G1 by A2;
    then A3: e in the_Edges_of G2 by Def10;
    per cases by A2, GLIB_000:61;
    suppose (the_Source_of G1).e = v;
      then (the_Source_of G2).e = v by Def10;
      then v in the_Vertices_of G2 by A3, FUNCT_2:5;
      hence contradiction by A1, XBOOLE_0:def 5;
    end;
    suppose (the_Target_of G1).e = v;
      then (the_Target_of G2).e = v by Def10;
      then v in the_Vertices_of G2 by A3, FUNCT_2:5;
      hence contradiction by A1, XBOOLE_0:def 5;
    end;
  end;
  hence v is isolated by GLIB_000:def 49;
  hence v is non cut-vertex;
end;
