
theorem
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for X,Y being Subset of the_Vertices_of G1 st F is weak_SG-embedding onto
  holds F_E.:G1.edgesBetween(X,Y) = G2.edgesBetween(F_V.:X,F_V.:Y)
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  let X,Y be Subset of the_Vertices_of G1;
  assume A1: F is weak_SG-embedding onto;
  then reconsider F9 = F as one-to-one PGraphMapping of G1, G2;
  A2: F_E.:G1.edgesBetween(X,Y) c= G2.edgesBetween(F_V.:X,F_V.:Y) by A1, Th91;
  A3: dom F_V = the_Vertices_of G1 & dom F_E = the_Edges_of G1
    by A1, GLIB_010:def 11;
  G1.edgesBetween(F9"_V.:(F_V.:X),F9"_V.:(F_V.:Y))
     = G1.edgesBetween(F9_V"(F9_V.:X),F9_V".:(F_V.:Y)) by FUNCT_1:85
    .= G1.edgesBetween(F9_V"(F9_V.:X),F9_V"(F_V.:Y)) by FUNCT_1:85
    .= G1.edgesBetween(X,F9_V"(F_V.:Y)) by A3, FUNCT_1:94
    .= G1.edgesBetween(X,Y) by A3, FUNCT_1:94;
  then A4: F9"_E.:G2.edgesBetween(F_V.:X,F_V.:Y) c= G1.edgesBetween(X,Y)
    by A1, GLIB_010:72, Th91;
  A5: rng F_E = the_Edges_of G2 by A1, GLIB_010:def 12;
  F_E.:(F9"_E.:G2.edgesBetween(F_V.:X,F_V.:Y))
     = F_E.:(F9_E"G2.edgesBetween(F_V.:X,F_V.:Y)) by FUNCT_1:85
    .= G2.edgesBetween(F_V.:X,F_V.:Y) by A5, FUNCT_1:77;
  then G2.edgesBetween(F_V.:X,F_V.:Y) c= F_E.:(G1.edgesBetween(X,Y))
    by A4, RELAT_1:123;
  hence thesis by A2, XBOOLE_0:def 10;
end;
