reserve G,G1,G2 for _Graph;
reserve W,W1,W2 for Walk of G;
reserve e,x,y,z for set;
reserve v for Vertex of G;
reserve n,m for Element of NAT;

theorem Th174:
  for G1 being _Graph, G2 being Subgraph of G1, W1 being Walk of
G1, W2 being Walk of G2 st W1 = W2 holds (W1 is closed iff W2 is closed) & (W1
  is directed iff W2 is directed) & (W1 is trivial iff W2 is trivial) & (W1 is
Trail-like iff W2 is Trail-like) & (W1 is Path-like iff W2 is Path-like) & (W1
  is vertex-distinct iff W2 is vertex-distinct)
proof
  let G1 be _Graph, G2 be Subgraph of G1, W1 be Walk of G1, W2 be Walk of G2;
  assume
A1: W1 = W2;
  then
A2: W1.last() = W2.last();
  W1.first() = W2.first() by A1;
  hence W1 is closed iff W2 is closed by A2;
  now
    hereby
      assume
A3:   W1 is directed;
      now
        let n be odd Element of NAT;
A4:     1 <= n+1 by NAT_1:12;
        assume
A5:     n < len W2;
        then n+1 <= len W2 by NAT_1:13;
        then n+1 in dom W2 by A4,FINSEQ_3:25;
        then W2.(n+1) in the_Edges_of G2 by Th7;
        then
        (the_Source_of G2).(W2.(n+1)) = (the_Source_of G1).(W2.(n +1)) by
GLIB_000:def 32;
        hence (the_Source_of G2).(W2.(n+1)) = W2.n by A1,A3,A5;
      end;
      hence W2 is directed;
    end;
    assume W2 is directed;
    hence W1 is directed Walk of G1 by A1,Th173;
  end;
  hence W1 is directed iff W2 is directed;
  W1 is trivial iff len W2 = 1 by A1,Lm55;
  hence W1 is trivial iff W2 is trivial by Lm55;
  W1 is Trail-like iff for m,n being even Element of NAT st 1 <= m & m <
  n & n <= len W2 holds W2.m <> W2.n by A1,Lm57;
  hence
A6: W1 is Trail-like iff W2 is Trail-like by Lm57;
  W1 is Path-like iff (W1 is Trail-like & for m,n being odd Element of
NAT st m < n & n <= len W2 holds W2.m = W2.n implies m = 1 & n = len W2 ) by A1
;
  hence W1 is Path-like iff W2 is Path-like by A6;
  W1 is vertex-distinct iff for m,n being odd Element of NAT st m <= len
  W2 & n <= len W2 & W2.m = W2.n holds m = n by A1;
  hence thesis;
end;
