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
  for G1 being _Graph, W being Walk of G1, e being set, G2 being
  removeEdge of G1,e st not e in W.edges() holds W is Walk of G2
proof
  let G1 be _Graph, W be Walk of G1, e be set, G2 be removeEdge of G1,e;
A1: the_Edges_of G2 = (the_Edges_of G1) \ {e} by GLIB_000:53;
  assume
A2: not e in W.edges();
  now
    let x be object;
    assume
A3: x in W.edges();
    then not x in {e} by A2,TARSKI:def 1;
    hence x in the_Edges_of G2 by A1,A3,XBOOLE_0:def 5;
  end;
  then
A4: W.edges() c= the_Edges_of G2 by TARSKI:def 3;
  the_Vertices_of G2 = the_Vertices_of G1 by GLIB_000:53;
  then W.vertices() c= the_Vertices_of G2;
  hence thesis by A4,Th168;
end;
