
theorem
  for G1 being non-Dmulti _Graph, G2 be _Graph
  for F being directed PGraphMapping of G1, G2
  st F_V is one-to-one holds F_E is one-to-one
proof
  let G1 be non-Dmulti _Graph, G2 be _Graph;
  let F be directed PGraphMapping of G1, G2;
  assume A1: F_V is one-to-one;
  now
    let e1, e2 be object;
    set v1 = (the_Source_of G1).e1, w1 = (the_Target_of G1).e1,
      v2 = (the_Source_of G1).e2, w2 = (the_Target_of G1).e2;
    assume A2: e1 in dom F_E & e2 in dom F_E & F_E.e1 = F_E.e2;
    then A3: v1 in dom F_V & w1 in dom F_V & v2 in dom F_V & w2 in dom F_V
      by Th5;
    A4: e1 DJoins v1,w1,G1 & e2 DJoins v2,w2,G1 by A2, GLIB_000:def 14;
    then F_E.e1 DJoins F_V.v1,F_V.w1,G2 & F_E.e2 DJoins F_V.v2,F_V.w2,G2
      by A2, A3, Def14;
    then F_V.v1 = F_V.v2 & F_V.w1 = F_V.w2 by A2, GLIB_000:125;
    then v1 = v2 & w1 = w2 by A1, A3, FUNCT_1:def 4;
    hence e1 = e2 by A4, GLIB_000:def 21;
  end;
  hence thesis by FUNCT_1:def 4;
end;
