
theorem Th16:
  for L being non empty RelStr, N being constant net of L, M being
  subnet of N holds M is constant & the_value_of N = the_value_of M
proof
  let L be non empty RelStr, N be constant net of L, M be subnet of N;
  consider f being Function of M, N such that
A1: the mapping of M = (the mapping of N)*f and
  for m being Element of N ex n being Element of M st for p being Element
  of M st n <= p holds m <= f.p by YELLOW_6:def 9;
  set y = the Element of dom (the mapping of M);
A2: dom (the mapping of N) = the carrier of N by FUNCT_2:def 1;
  then
A3: the_value_of the mapping of N= (the mapping of N).(f.y) by FUNCT_1:def 12
    .= (the mapping of M).y by A1,FUNCT_1:12;
A4: dom f = the carrier of M by FUNCT_2:def 1;
  for n1,n2 being object st n1 in dom (the mapping of M) & n2 in dom (the
  mapping of M) holds (the mapping of M).n1=(the mapping of M).n2
  proof
    let n1,n2 be object;
    assume that
A5: n1 in dom (the mapping of M) and
A6: n2 in dom (the mapping of M);
A7: f.n1 in rng f & f.n2 in rng f by A4,A5,A6,FUNCT_1:def 3;
    thus (the mapping of M).n1= (the mapping of N).(f.n1) by A1,A4,A5,
FUNCT_1:13
      .= (the mapping of N).(f.n2) by A2,A7,FUNCT_1:def 10
      .= (the mapping of M).n2 by A1,A4,A6,FUNCT_1:13;
  end;
  then
A8: the mapping of M is constant by FUNCT_1:def 10;
  hence
A9: M is constant;
  thus the_value_of N = the_value_of the mapping of N by YELLOW_6:def 8
    .= the_value_of the mapping of M by A8,A3,FUNCT_1:def 12
    .= the_value_of M by A9,YELLOW_6:def 8;
end;
