
theorem Th48:
  for G being _Graph, v being Vertex of G, W being Walk of G
  st W is_Walk_from v,v holds W is minlength iff W = G.walkOf(v)
proof
  let G be _Graph, v be Vertex of G, W be Walk of G;
  assume A1: W is_Walk_from v,v;
  hereby
    assume A2: W is minlength;
    G.walkOf(v) is_Walk_from v,v by GLIB_001:13;
    then G.walkOf(v) is_Walk_from W.first(),v by A1, GLIB_001:def 23;
    then G.walkOf(v) is_Walk_from W.first(),W.last() by A1, GLIB_001:def 23;
    then len G.walkOf(v) >= len W by A2, CHORD:def 2;
    then A3: 1 >= len W by GLIB_001:13;
    1 <= len W by CHORD:2;
    then W is trivial by A3, XXREAL_0:1, GLIB_001:126;
    then consider v0 being Vertex of G such that
      A4: W = G.walkOf(v0) by GLIB_001:128;
    W.first() = v0 by A4;
    hence W = G.walkOf(v) by A1, A4, GLIB_001:def 23;
  end;
  assume A5: W = G.walkOf(v);
  now
    let W9 be Walk of G;
    assume W9 is_Walk_from W.first(),W.last();
    len W9 >= 1 by CHORD:2;
    hence len W9 >= len W by A5, GLIB_001:13;
  end;
  hence thesis by CHORD:def 2;
end;
