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 Th173:
  for G1 being _Graph, G2 being Subgraph of G1, W being Walk of
G2 holds (W is closed implies W is closed Walk of G1) & (W is directed implies
W is directed Walk of G1) & (W is trivial implies W is trivial Walk of G1) & (W
  is Trail-like implies W is Trail-like Walk of G1) & (W is Path-like implies W
  is Path-like Walk of G1) & (W is vertex-distinct implies W is vertex-distinct
  Walk of G1)
proof
  let G1 be _Graph, G2 be Subgraph of G1, W be Walk of G2;
  reconsider W9=W as Walk of G1 by Th165;
  hereby
    assume W is closed;
    then W.first() = W.last();
    then W9.first() = W9.last();
    hence W is closed Walk of G1 by Def24;
  end;
  hereby
    assume
A1: W is directed;
    now
      let n be odd Element of NAT;
A2:   1 <= n+1 by NAT_1:12;
      assume
A3:   n < len W9;
      then n+1 <= len W9 by NAT_1:13;
      then n+1 in dom W9 by A2,FINSEQ_3:25;
      then
A4:   W9.(n+1) in the_Edges_of G2 by Th7;
      (the_Source_of G2).(W9.(n+1)) = W9.n by A1,A3;
      hence (the_Source_of G1).(W9.(n+1)) = W9.n by A4,GLIB_000:def 32;
    end;
    hence W is directed Walk of G1 by Def25;
  end;
  hereby
    assume W is trivial;
    then len W9 = 1 by Lm55;
    hence W is trivial Walk of G1 by Lm54;
  end;
A5: now
    assume W is Trail-like;
    then for m,n being even Element of NAT st 1 <= m & m < n & n <= len W
    holds W9.m <> W9.n by Lm57;
    hence W is Trail-like Walk of G1 by Lm57;
  end;
  hence W is Trail-like implies W is Trail-like Walk of G1;
  W is Path-like implies W is Path-like Walk of G1 by A5,Def28;
  hence W is Path-like implies W is Path-like Walk of G1;
  hereby
    assume W is vertex-distinct;
    then for m,n being odd Element of NAT st m <= len W9 & n <= len W9 & W9.m
    = W9.n holds m = n;
    hence W is vertex-distinct Walk of G1 by Def29;
  end;
end;
