
theorem
  for G2, G3 being _Graph, v,e,w being object
  for G1 being addAdjVertex of G3,v,e,w st not e in the_Edges_of G3 holds
    G2 is reverseEdgeDirections of G1, {e} iff G2 is addAdjVertex of G3,w,e,v
proof
  let G2, G3 be _Graph, v,e,w be object;
  let G1 be addAdjVertex of G3,v,e,w;
  assume A1: not e in the_Edges_of G3;
  per cases;
  suppose A2: v in the_Vertices_of G3 & not w in the_Vertices_of G3;
    then consider G4 being addVertex of G3,w such that
      A3: G1 is addEdge of G4,v,e,w by A1, GLIB_006:125;
    A4: the_Edges_of G4 = the_Edges_of G3 by GLIB_006:def 10;
    hereby
      assume G2 is reverseEdgeDirections of G1, {e};
      then G2 is addEdge of G4,w,e,v by A1, A3, A4, Th66;
      hence G2 is addAdjVertex of G3,w,e,v by A1, A2, GLIB_006:128;
    end;
    assume G2 is addAdjVertex of G3,w,e,v;
    then consider G5 being addVertex of G3,w such that
      A5: G2 is addEdge of G5,w,e,v by A1, A2, GLIB_006:126;
    G2 is addEdge of G4,w,e,v by A5, GLIB_006:77, GLIB_008:36;
    hence G2 is reverseEdgeDirections of G1, {e} by A1, A3, A4, Th66;
  end;
  suppose A6: not v in the_Vertices_of G3 & w in the_Vertices_of G3;
    then consider G4 being addVertex of G3,v such that
      A7: G1 is addEdge of G4,v,e,w by A1, GLIB_006:126;
    A8: the_Edges_of G4 = the_Edges_of G3 by GLIB_006:def 10;
    hereby
      assume G2 is reverseEdgeDirections of G1, {e};
      then G2 is addEdge of G4,w,e,v by A1, A7, A8, Th66;
      hence G2 is addAdjVertex of G3,w,e,v by A1, A6, GLIB_006:127;
    end;
    assume G2 is addAdjVertex of G3,w,e,v;
    then consider G5 being addVertex of G3,v such that
      A9: G2 is addEdge of G5,w,e,v by A1, A6, GLIB_006:125;
    G2 is addEdge of G4,w,e,v by A9, GLIB_006:77, GLIB_008:36;
    hence G2 is reverseEdgeDirections of G1, {e} by A1, A7, A8, Th66;
  end;
  suppose A10: not((v in the_Vertices_of G3 & not w in the_Vertices_of G3) or
      (not v in the_Vertices_of G3 & w in the_Vertices_of G3));
    then A11: G1 == G3 by GLIB_006:def 12;
    then the_Edges_of G1 = the_Edges_of G3 by GLIB_000:def 34;
    then A12: not {e} c= the_Edges_of G1 by A1, ZFMISC_1:31;
    hereby
      assume G2 is reverseEdgeDirections of G1, {e};
      then G2 == G1 by A12, GLIB_007:def 1;
      then A13: G2 == G3 by A11, GLIB_000:85;
      then G2 is Supergraph of G3 by GLIB_006:59;
      hence G2 is addAdjVertex of G3,w,e,v by A10, A13, GLIB_006:def 12;
    end;
    assume G2 is addAdjVertex of G3,w,e,v;
    then G2 == G3 by A10, GLIB_006:def 12;
    then G2 == G1 by A11, GLIB_000:85;
    hence thesis by A12, GLIB_007:def 1;
  end;
end;
