reserve G,G1,G2 for _Graph;
reserve e,x,y for set;
reserve v,v1,v2 for Vertex of G;
reserve W for Walk of G;

theorem
  for v1 being Vertex of G1, v2 being Vertex of G2 st G1 == G2 & v1 = v2
  holds G1.reachableDFrom(v1) = G2.reachableDFrom(v2)
proof
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume that
A1: G1 == G2 and
A2: v1 = v2;
  now
    let x be object;
    hereby
      assume x in G1.reachableDFrom(v1);
      then consider W being DWalk of G1 such that
A3:   W is_Walk_from v2,x by A2,Def6;
      reconsider W2 = W as DWalk of G2 by A1,GLIB_001:179,181;
      W2 is_Walk_from v2,x by A3,GLIB_001:19;
      hence x in G2.reachableDFrom(v2) by Def6;
    end;
    assume x in G2.reachableDFrom(v2);
    then consider W being DWalk of G2 such that
A4: W is_Walk_from v1,x by A2,Def6;
    reconsider W2 = W as DWalk of G1 by A1,GLIB_001:179,181;
    W2 is_Walk_from v1,x by A4,GLIB_001:19;
    hence x in G1.reachableDFrom(v1) by Def6;
  end;
  hence thesis by TARSKI:2;
end;
