reserve G for _Graph;
reserve V for non empty set, E for Relation of V;
reserve E for symmetric Relation of V;
reserve G for GraphFromSymRel of V, E;

theorem
  for G being GraphFromSymRel of V, E holds VertexAdjSymRel(G) = E
proof
  let G be GraphFromSymRel of V, E;
  now
    let x,y be object;
    hereby
      assume A1: [x,y] in VertexAdjSymRel(G);
      then reconsider v = x, w = y as Vertex of G by ZFMISC_1:87;
      v,w are_adjacent by A1, Th33;
      hence [x,y] in E by Th89;
    end;
    set G0 = createGraph(V,E);
    consider E0 being RepEdgeSelection of G0 such that
      A2: G is inducedSubgraph of G0, the_Vertices_of G0, E0 by GLIB_009:def 7;
    A3: the_Edges_of G0 = G0.edgesBetween(the_Vertices_of G0) by GLIB_000:34;
    the_Vertices_of G0 c= the_Vertices_of G0;
    then A4: the_Vertices_of G = the_Vertices_of G0 by A2, A3, GLIB_000:def 37
      .= V;
    assume A5: [x,y] in E;
    then reconsider v = x, w = y as Vertex of G by A4, ZFMISC_1:87;
    v,w are_adjacent by A5, Th89;
    hence [x,y] in VertexAdjSymRel(G) by Th33;
  end;
  hence thesis by RELAT_1:def 2;
end;
