
theorem
  for G1 being non-multi _Graph, G2 being _Graph
  for F being semi-continuous PGraphMapping of G1, G2
  holds F_E is one-to-one
proof
  let G1 be non-multi _Graph, G2 be _Graph;
  let F be semi-continuous PGraphMapping of G1, G2;
  now
    let e1, e2 be object;
    assume A1: e1 in dom F_E & e2 in dom F_E & F_E.e1 = F_E.e2;
    then A2: (the_Source_of G1).e1 in dom F_V &
      (the_Target_of G1).e1 in dom F_V by Th5;
    A3: e1 Joins (the_Source_of G1).e1, (the_Target_of G1).e1, G1
      by A1, GLIB_000:def 13;
    then F_E.e2 Joins F_V.((the_Source_of G1).e1),
      F_V.((the_Target_of G1).e1), G2 by A1, A2, Th4;
    then e2 Joins (the_Source_of G1).e1, (the_Target_of G1).e1, G1
      by A1, A2, Def15;
    hence e1 = e2 by A3, GLIB_000:def 20;
  end;
  hence thesis by FUNCT_1:def 4;
end;
