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 Th89:
  for v, w being Vertex of G holds [v,w] in E iff v,w are_adjacent
proof
  let v, w be Vertex of G;
  hereby
    assume [v,w] in E;
    then per cases by Th88;
    suppose [v,w] DJoins v,w,G;
      then [v,w] Joins v,w,G by GLIB_000:16;
      hence v,w are_adjacent by CHORD:def 3;
    end;
    suppose [w,v] DJoins w,v,G;
      then [w,v] Joins v,w,G by GLIB_000:16;
      hence v,w are_adjacent by CHORD:def 3;
    end;
  end;
  set G0 = createGraph(V,E);
  consider E0 being RepEdgeSelection of G0 such that
    A1: G is inducedSubgraph of G0, the_Vertices_of G0, E0 by GLIB_009:def 7;
  A2: the_Edges_of G0 = G0.edgesBetween(the_Vertices_of G0) by GLIB_000:34;
  the_Vertices_of G0 c= the_Vertices_of G0;
  then A3: the_Edges_of G = E0 by A1, A2, GLIB_000:def 37;
  assume v,w are_adjacent;
  then consider e being object such that
    A4: e Joins v,w,G by CHORD:def 3;
  e in E0 by A3, A4, GLIB_000:def 13;
  then e in the_Edges_of G0;
  then A5: e in E;
  then consider v0, w0 being object such that
    A6: e = [v0,w0] by RELAT_1:def 1;
  e DJoins v0,w0,G0 by A5, A6, Th63;
  then A7: e Joins v0,w0,G0 by GLIB_000:16;
  e Joins v,w,G0 by A4, GLIB_000:72;
  then per cases by A7, GLIB_000:15;
  suppose v = v0 & w = w0;
    hence [v,w] in E by A5, A6;
  end;
  suppose v = w0 & w = v0;
    hence [v,w] in E by A5, A6, GLIBPRE0:13;
  end;
end;
