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
  G1 == G2 & G1 is_DTree_rooted_at x implies G2 is_DTree_rooted_at x
proof
  assume that
A1: G1 == G2 and
A2: G1 is_DTree_rooted_at x;
A3: now
    let y be Vertex of G2;
    reconsider y9 = y as Vertex of G1 by A1,GLIB_000:def 34;
    consider W being DWalk of G1 such that
A4: W is_Walk_from x,y9 by A2;
    reconsider W9=W as DWalk of G2 by A1,GLIB_001:179,181;
    take W9;
    thus W9 is_Walk_from x,y by A4,GLIB_001:19;
  end;
  G1 is Tree-like by A2;
  then G2 is Tree-like by A1,Th47;
  hence thesis by A3;
end;
