
theorem Th91:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for X,Y being Subset of the_Vertices_of G1 st F is total
  holds F_E.:G1.edgesBetween(X,Y) c= 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 total;
  set f = F_E | G1.edgesBetween(X,Y);
  A2: dom f = dom F_E /\ G1.edgesBetween(X,Y) by RELAT_1:61
    .= the_Edges_of G1 /\ G1.edgesBetween(X,Y) by A1, GLIB_010:def 11
    .= G1.edgesBetween(X,Y) by XBOOLE_1:28;
  for y being object holds y in rng f implies
    y in G2.edgesBetween(F_V.:X,F_V.:Y)
  proof
    let y be object;
    assume y in rng f;
    then consider x being object such that
      A3: x in dom f & f.x = y by FUNCT_1:def 3;
    set v = (the_Source_of G1).x, w = (the_Target_of G1).x;
    A4: x SJoins X,Y,G1 by A2, A3, GLIB_000:def 30;
    then per cases by GLIB_000:def 15;
    suppose A5: v in X & w in Y;
      then v in the_Vertices_of G1 & w in the_Vertices_of G1;
      then A6: v in dom F_V & w in dom F_V by A1, GLIB_010:def 11;
      A7: x in the_Edges_of G1 by A4, GLIB_000:def 15;
      then A8: x in dom F_E by A1, GLIB_010:def 11;
      x Joins v,w,G1 by A7, GLIB_000:def 13;
      then F_E.x Joins F_V.v,F_V.w,G2 by A6, A8, GLIB_010:4;
      then A9: y Joins F_V.v,F_V.w,G2 by A3, FUNCT_1:47;
      F_V.v in F_V.:X & F_V.w in F_V.:Y by A5, A6, FUNCT_1:def 6;
      then y SJoins F_V.:X,F_V.:Y,G2 by A9, GLIB_000:17;
      hence thesis by GLIB_000:def 30;
    end;
    suppose A10: v in Y & w in X;
      then v in the_Vertices_of G1 & w in the_Vertices_of G1;
      then A11: v in dom F_V & w in dom F_V by A1, GLIB_010:def 11;
      A12: x in the_Edges_of G1 by A4, GLIB_000:def 15;
      then A13: x in dom F_E by A1, GLIB_010:def 11;
      x Joins v,w,G1 by A12, GLIB_000:def 13;
      then F_E.x Joins F_V.v,F_V.w,G2 by A11, A13, GLIB_010:4;
      then A14: y Joins F_V.v,F_V.w,G2 by A3, FUNCT_1:47;
      F_V.v in F_V.:Y & F_V.w in F_V.:X by A10, A11, FUNCT_1:def 6;
      then y SJoins F_V.:X,F_V.:Y,G2 by A14, GLIB_000:17;
      hence thesis by GLIB_000:def 30;
    end;
  end;
  then rng f c= G2.edgesBetween(F_V.:X,F_V.:Y) by TARSKI:def 3;
  hence thesis by RELAT_1:115;
end;
