
theorem Th26:
  for G1, G2 being _Graph
  for F being semi-continuous PGraphMapping of G1, G2, v1, v2 being object
  st v1 in dom F_V & v2 in dom F_V & F_V.v1 = F_V.v2 &
    ex e, w being object
    st e in dom F_E & w in dom F_V & F_E.e Joins F_V.v1, F_V.w, G2
  holds v1 = v2
proof
  let G1, G2 be _Graph, F be semi-continuous PGraphMapping of G1, G2;
  let v1, v2 be object;
  assume that
    A1: v1 in dom F_V & v2 in dom F_V & F_V.v1 = F_V.v2 and
    A2: ex e, w being object st e in dom F_E & w in dom F_V &
      F_E.e Joins F_V.v1, F_V.w, G2;
  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 A2;
  A4: e Joins v1,w,G1 by A1, A3, Def15;
  e Joins v2,w,G1 by A1, A3, Def15;
  then per cases by A4, GLIB_000:15;
  suppose v1 = v2 & w = w;
    hence v1 = v2;
  end;
  suppose v1 = w & w = v2;
    hence v1 = v2;
  end;
end;
