reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;
reserve V for set;

theorem Th94:
  for G2, V for G1 being addVertices of G2, V, W being Walk of G1
  holds W.vertices() misses V \ the_Vertices_of G2 or W is trivial
proof
  let G2, V;
  let G1 be addVertices of G2, V, W be Walk of G1;
  assume not W.vertices() misses V \ the_Vertices_of G2;
  then W.vertices() /\ (V \ the_Vertices_of G2) <> {} by XBOOLE_0:def 7;
  then consider v being object such that
    A1: v in W.vertices() /\ (V \ the_Vertices_of G2) by XBOOLE_0:def 1;
  v in W.vertices() & v in V \ the_Vertices_of G2 by A1, XBOOLE_0:def 4;
  then A2: v in W.vertices() & v in V & not v in the_Vertices_of G2
    by XBOOLE_0:def 5;
  then consider n being odd Element of NAT such that
    A3: n <= len W & W.n = v by GLIB_001:87;
  per cases by A3, XXREAL_0:1;
  suppose n < len W;
    then per cases by GLIB_001:def 3, GLIB_000:16;
    suppose W.(n+1) DJoins W.n,W.(n+2),G1;
      then W.(n+1) DJoins W.n,W.(n+2),G2 by Th89;
      then W.(n+1) in the_Edges_of G2 &
        (the_Source_of G2).(W.(n+1)) = W.n by GLIB_000:def 14;
      hence thesis by A2, A3, FUNCT_2:5;
    end;
    suppose W.(n+1) DJoins W.(n+2),W.n,G1;
      then W.(n+1) DJoins W.(n+2),W.n,G2 by Th89;
      then W.(n+1) in the_Edges_of G2 &
        (the_Target_of G2).(W.(n+1)) = W.n by GLIB_000:def 14;
      hence thesis by A2, A3, FUNCT_2:5;
    end;
  end;
  suppose A4: n = len W;
    per cases;
    suppose n = 1;
      hence thesis by A4, GLIB_001:126;
    end;
    suppose n <> 1;
      then consider m being odd Nat such that
        A5: m+2 = n by CHORD:5;
      reconsider m as odd Element of NAT by ORDINAL1:def 12;
      m < len W by A4, A5, NAT_D:47;
      then per cases by GLIB_000:16, GLIB_001:def 3;
      suppose W.(m+1) DJoins W.m,W.(m+2),G1;
        then W.(m+1) DJoins W.m,W.(m+2),G2 by Th89;
        then W.(m+1) in the_Edges_of G2 &
          (the_Target_of G2).(W.(m+1)) = W.(m+2) by GLIB_000:def 14;
        hence thesis by A2, A3, A5, FUNCT_2:5;
      end;
      suppose W.(m+1) DJoins W.(m+2),W.m,G1;
        then W.(m+1) DJoins W.(m+2),W.m,G2 by Th89;
        then W.(m+1) in the_Edges_of G2 &
          (the_Source_of G2).(W.(m+1)) = W.(m+2) by GLIB_000:def 14;
        hence thesis by A2, A3, A5, FUNCT_2:5;
      end;
    end;
  end;
end;
