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
  G1 == G2 implies G1.walkOf(x,e,y) = G2.walkOf(x,e,y)
proof
  assume
A1: G1 == G2;
  now
    per cases;
    suppose
A2:   e Joins x,y,G1;
      then
A3:   e Joins x,y,G2 by A1,GLIB_000:88;
      thus G1.walkOf(x,e,y) = <*x,e,y*> by A2,Def5
        .= G2.walkOf(x,e,y) by A3,Def5;
    end;
    suppose
A4:   not e Joins x,y,G1;
      then
A5:   not e Joins x,y,G2 by A1,GLIB_000:88;
A6:     the_Vertices_of G1 = the_Vertices_of G2 by A1,GLIB_000:def 34;
      thus G1.walkOf(x,e,y) =
          G1.walkOf(the Element of the_Vertices_of G1) by A4,Def5
        .= G2.walkOf(the Element of the_Vertices_of G2)
                 by A6
        .= G2.walkOf(x,e,y) by A5,Def5;
    end;
  end;
  hence thesis;
end;
