
theorem Th20:
  for S, T being complete LATTICE, f being Function of S, T, N being net of S,
  j being Element of N, j9 being Element of (f*N) st j9 = j
  holds f is monotone implies f."/\"({N.k where k is Element of N: k >= j},S)
  <= "/\"({(f*N).l where l is Element of (f*N) : l >= j9},T)
proof
  let S, T be complete LATTICE, f be Function of S, T;
  let N be net of S;
  let j be Element of N, j9 be Element of (f*N);
  assume
A1: j9 = j;
  assume
A2: f is monotone;
A3: dom f = the carrier of S by FUNCT_2:def 1;
A4: the RelStr of (f*N) = the RelStr of N by WAYBEL_9:def 8;
A5: dom (the mapping of N) = the carrier of N by FUNCT_2:def 1;
A6: {(f*N).l where l is Element of (f*N) : l >= j9} c=
  f.:{N.l1 where l1 is Element of N : l1 >= j}
  proof
    let s be object;
    assume s in {(f*N).l where l is Element of (f*N) : l >= j9};
    then consider x being Element of (f*N) such that
A7: s = (f*N).x and
A8: x >= j9;
    reconsider x as Element of N by A4;
    [j9,x] in the InternalRel of (f*N) by A8,ORDERS_2:def 5;
    then
A9: x >= j by A1,A4,ORDERS_2:def 5;
A10: s = (f*the mapping of N).x by A7,WAYBEL_9:def 8
      .= f.(N.x) by A5,FUNCT_1:13;
    N.x in {N.z where z is Element of N : z >= j} by A9;
    hence thesis by A3,A10,FUNCT_1:def 6;
  end;
A11: f.:{N.l1 where l1 is Element of N : l1 >= j} c=
  {(f*N).l where l is Element of (f*N) : l >= j9}
  proof
    let s be object;
    assume s in f.:{N.l1 where l1 is Element of N : l1 >= j};
    then consider x be object such that
    x in dom f and
A12: x in {N.l1 where l1 is Element of N : l1 >= j} and
A13: s = f.x by FUNCT_1:def 6;
    consider l2 being Element of N such that
A14: x = N.l2 and
A15: l2 >= j by A12;
    reconsider l29 = l2 as Element of (f*N) by A4;
A16: s = (f*the mapping of N).l2 by A5,A13,A14,FUNCT_1:13
      .= (f*N).l29 by WAYBEL_9:def 8;
    [j, l2] in the InternalRel of N by A15,ORDERS_2:def 5;
    then l29 >= j9 by A1,A4,ORDERS_2:def 5;
    hence thesis by A16;
  end;
  set K = {N.k where k is Element of N: k >= j};
  K c= the carrier of S
  proof
    let r be object;
    assume r in K;
    then ex f being Element of N st ( r = N.f)&( f >= j);
    hence thesis;
  end;
  then reconsider K as Subset of S;
  {(f*N).l where l is Element of (f*N) : l >= j9} = f.:K by A6,A11;
  hence thesis by A2,Lm7;
end;
