reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;
reserve V for set;
reserve v for object;

theorem Th113:
  for G2 for v1, v2 being Vertex of G2, e being object
  for G1 being addEdge of G2,v1,e,v2, W being Walk of G1
  st (e in W.edges() implies e in the_Edges_of G2)
  holds W is Walk of G2
proof
  let G2;
  let v1, v2 be Vertex of G2;
  let e be object;
  let G1 be addEdge of G2,v1,e,v2;
  let W be Walk of G1;
  assume e in W.edges() implies e in the_Edges_of G2;
  then per cases;
  suppose not e in W.edges();
    then A1: W.edges() misses {e} by ZFMISC_1:50;
    A2: G2 is Subgraph of G1 by Th61;
    per cases;
    suppose A3: not e in the_Edges_of G2;
      W.edges() c= the_Edges_of G1;
      then W.edges() c= the_Edges_of G2 \/ {e} by A3, Def11;
      then A4: W.edges() c= the_Edges_of G2 by A1, XBOOLE_1:73;
      W.vertices() c= the_Vertices_of G1;
      then W.vertices() c= the_Vertices_of G2 by A3, Def11;
      hence thesis by A2, A4, GLIB_001:170;
    end;
    suppose e in the_Edges_of G2;
      then G1 == G2 by Def11;
      hence thesis by GLIB_001:179;
    end;
  end;
  suppose e in the_Edges_of G2;
    then G1 == G2 by Def11;
    hence thesis by GLIB_001:179;
  end;
end;
