
theorem
  for G1 being _Graph, G2 being SimpleGraph of G1
  for v1,w1 being Vertex of G1, v2,w2 being Vertex of G2
  st v1 = v2 & w1 = w2 & v1 <> w1
  holds v1,w1 are_adjacent iff v2,w2 are_adjacent
proof
  let G1 be _Graph, G2 be SimpleGraph of G1;
  let v1,w1 be Vertex of G1, v2,w2 be Vertex of G2;
  assume A1: v1 = v2 & w1 = w2 & v1 <> w1;
  consider G9 being removeParallelEdges of G1 such that
    A2: G2 is removeLoops of G9 by GLIB_009:119;
  reconsider v9 = v2, w9 = w2 as Vertex of G9 by A2, GLIB_000:def 33;
  v1,w1 are_adjacent iff v9,w9 are_adjacent by A1, Th76;
  hence thesis by A1, A2, Th75;
end;
