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 Th166:
  for G1 being _Graph, G2 being Subgraph of G1, W being Walk of G1 st
  W is trivial & W.first() in the_Vertices_of G2 holds W is Walk of G2
proof
  let G1 be _Graph, G2 be Subgraph of G1, W be Walk of G1;
  assume that
A1: W is trivial and
A2: W.first() in the_Vertices_of G2;
  consider v being Vertex of G1 such that
A3: W = G1.walkOf(v) by A1,Lm56;
  reconsider v9= v as Vertex of G2 by A2,A3;
  W = G2.walkOf(v9) by A3;
  hence thesis;
end;
