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 Th165:
  for G1 being _Graph, G2 being Subgraph of G1, W being Walk of G2 holds
  W is Walk of G1
proof
  let G1 be _Graph, G2 be Subgraph of G1, W be Walk of G2;
  set VG1 = the_Vertices_of G1, VG2 = the_Vertices_of G2;
  set EG1 = the_Edges_of G1, EG2 = the_Edges_of G2;
A1: EG2 c= VG1 \/ EG1 by XBOOLE_1:10;
A2: now
    thus len W is odd;
    W.1 in VG2 by Def3;
    hence W.1 in VG1;
    let n be odd Element of NAT;
    assume n < len W;
    then W.(n+1) Joins W.n, W.(n+2), G2 by Def3;
    hence W.(n+1) Joins W.n, W.(n+2), G1 by GLIB_000:72;
  end;
  VG2 c= VG1 \/ EG1 by XBOOLE_1:10;
  then VG2 \/ EG2 c= VG1 \/ EG1 by A1,XBOOLE_1:8;
  then for y being object st y in rng W holds y in VG1 \/ EG1 by TARSKI:def 3;
  then rng W c= VG1 \/ EG1 by TARSKI:def 3;
  then W is FinSequence of VG1 \/ EG1 by FINSEQ_1:def 4;
  hence thesis by A2,Def3;
end;
