
theorem Th104:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2, V being set
  st F is continuous holds F_E.:G1.edgesBetween(V) = G2.edgesBetween(F_V.:V)
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2, V be set;
  assume A1: F is continuous;
  A2: F_E.:G1.edgesBetween(V) c= G2.edgesBetween(F_V.:V) by Th103;
  now
    let e be object;
    set v = (the_Source_of G2).e, w = (the_Target_of G2).e;
    assume e in G2.edgesBetween(F_V.:V);
    then A3: e in the_Edges_of G2 & v in F_V.:V & w in F_V.:V by GLIB_000:31;
    then A4: e Joins v,w,G2 by GLIB_000:def 13;
    consider v0 being object such that
      A5: v0 in dom F_V & v0 in V & F_V.v0 = v by A3, FUNCT_1:def 6;
    consider w0 being object such that
      A6: w0 in dom F_V & w0 in V & F_V.w0 = w by A3, FUNCT_1:def 6;
    consider e0 being object such that
      A7: e0 Joins v0,w0,G1 & e0 in dom F_E & F_E.e0 = e
      by A1, A4, A5, A6, GLIB_010:def 16;
    e0 in G1.edgesBetween(V) by A5, A6, A7, GLIB_000:32;
    hence e in F_E.:G1.edgesBetween(V) by A7, FUNCT_1:def 6;
  end;
  then G2.edgesBetween(F_V.:V) c= F_E.:G1.edgesBetween(V) by TARSKI:def 3;
  hence thesis by A2, XBOOLE_0:def 10;
end;
