
theorem
  for G1 being non-multi _Graph, G2 being _Graph
  for F being PGraphMapping of G1, G2
  st F_V is one-to-one holds F_E is one-to-one
proof
  let G1 be non-multi _Graph, G2 be _Graph;
  let F be 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 Joins v1,w1,G1 & e2 Joins v2,w2,G1 by A2, GLIB_000:def 13;
    then F_E.e1 Joins F_V.v1,F_V.w1,G2 & F_E.e2 Joins F_V.v2,F_V.w2,G2
      by A2, A3, Th4;
    then per cases by A2, GLIB_000:15;
    suppose F_V.v1 = F_V.v2 & F_V.w1 = F_V.w2;
      then v1 = v2 & w1 = w2 by A1, A3, FUNCT_1:def 4;
      hence e1 = e2 by A4, GLIB_000:def 20;
    end;
    suppose F_V.v1 = F_V.w2 & F_V.w1 = F_V.v2;
      then v1 = w2 & w1 = v2 by A1, A3, FUNCT_1:def 4;
      then e2 Joins v1,w1,G1 by A4, GLIB_000:14;
      hence e1 = e2 by A4, GLIB_000:def 20;
    end;
  end;
  hence thesis by FUNCT_1:def 4;
end;
