
theorem
  for G1, G2 being _Graph
  for F being semi-continuous PGraphMapping of G1, G2
  st for v being object st v in dom F_V holds ex e, w being object
    st e in dom F_E & w in dom F_V & F_E.e Joins F_V.v, F_V.w, G2
  holds F_V is one-to-one
proof
  let G1, G2 be _Graph, F be semi-continuous PGraphMapping of G1, G2;
  assume A1: for v being object st v in dom F_V holds ex e, w being object
    st e in dom F_E & w in dom F_V & F_E.e Joins F_V.v, F_V.w, G2;
  now
    let v1, v2 be object;
    assume A2: v1 in dom F_V & v2 in dom F_V & F_V.v1 = F_V.v2;
    then consider e, w being object such that
      A3: e in dom F_E & w in dom F_V & F_E.e Joins F_V.v1, F_V.w, G2 by A1;
    thus v1 = v2 by A2, A3, Th26;
  end;
  hence thesis by FUNCT_1:def 4;
end;
