reserve G for _Graph;

theorem Th5:
  for G being loopless _Graph
  st field VertexDomRel(G) = the_Vertices_of G
  holds for C being Component of G holds C is non _trivial
proof
  let G be loopless _Graph;
  assume A1: field VertexDomRel(G) = the_Vertices_of G;
  let C be Component of G;
  assume C is _trivial;
  then consider v being Vertex of C such that
    A2: the_Vertices_of C = {v} by GLIB_000:22;
  A3: v in the_Vertices_of G by A2, ZFMISC_1:31;
  reconsider v0 = v as Vertex of G by A2, ZFMISC_1:31;
  the_Vertices_of G = dom VertexDomRel(G) \/ rng VertexDomRel(G)
    by A1, RELAT_1:def 6;
  then per cases by A3, XBOOLE_0:def 3;
  suppose v in dom VertexDomRel(G);
    then consider w being object such that
      A4: [v,w] in VertexDomRel(G) by XTUPLE_0:def 12;
    consider e being object such that
      A5: e DJoins v,w,G by A4, Th1;
    e Joins v,w,G by A5, GLIB_000:16;
    then w in G.reachableFrom(v0) by GLIB_002:9, GLIB_002:10;
    then w in the_Vertices_of C by GLIB_002:33;
    then w = v by A2, TARSKI:def 1;
    hence contradiction by A5, GLIB_000:136;
  end;
  suppose v in rng VertexDomRel(G);
    then consider w being object such that
      A6: [w,v] in VertexDomRel(G) by XTUPLE_0:def 13;
    consider e being object such that
      A7: e DJoins w,v,G by A6, Th1;
    e Joins v,w,G by A7, GLIB_000:16;
    then w in G.reachableFrom(v0) by GLIB_002:9, GLIB_002:10;
    then w in the_Vertices_of C by GLIB_002:33;
    then w = v by A2, TARSKI:def 1;
    hence contradiction by A7, GLIB_000:136;
  end;
end;
