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 Th88:
  for v,w being object
  holds [v,w] in E iff [v,w] DJoins v,w,G or [w,v] DJoins w,v,G
proof
  let v,w be object;
  A1: v is set & w is set by TARSKI:1;
  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;
  hereby
    assume [v,w] in E;
    then A3: [v,w] DJoins v,w,G0 & [w,v] DJoins w,v,G0 by Th63, GLIBPRE0:13;
    [v,w] Joins v,w,G0 by A3, GLIB_000:16;
    then consider e being object such that
      A4: e Joins v,w,G0 & e in E0 and
      for e9 being object st e9 Joins v,w,G0 & e9 in E0 holds e9 = e
      by GLIB_009:def 5;
    e in the_Edges_of G0 by A4;
    then A5: e in E;
    then consider v0, w0 being object such that
      A6: e = [v0,w0] by RELAT_1:def 1;
    A7: e DJoins v0,w0,G0 by A5, A6, Th63;
    A8: the_Edges_of G0 = G0.edgesBetween(the_Vertices_of G0) by GLIB_000:34;
    the_Vertices_of G0 c= the_Vertices_of G0;
    then A9: E0 = the_Edges_of G by A2, A8, GLIB_000:def 37;
    e is set & v0 is set & w0 is set by TARSKI:1;
    then A10: e DJoins v0,w0,G by A4, A7, A9, GLIB_000:73;
    e Joins v0,w0,G0 by A7, GLIB_000:16;
    then per cases by A4, GLIB_000:15;
    suppose v = v0 & w = w0;
      hence [v,w] DJoins v,w,G or [w,v] DJoins w,v,G by A6, A10;
    end;
    suppose v = w0 & w = v0;
      hence [v,w] DJoins v,w,G or [w,v] DJoins w,v,G by A6, A10;
    end;
  end;
  assume [v,w] DJoins v,w,G or [w,v] DJoins w,v,G;
  then per cases;
  suppose [v,w] DJoins v,w,G;
    hence [v,w] in E by A1, Th63, GLIB_000:72;
  end;
  suppose [w,v] DJoins w,v,G;
    then [w,v] DJoins w,v,G0 by A1, GLIB_000:72;
    hence [v,w] in E by Th63, GLIBPRE0:13;
  end;
end;
