
theorem

:: Proposition 2.15, p. 63
:: That Image p is infs-inheriting follows from O-3.11 (iii)
  for L being continuous complete LATTICE, p being projection Function
  of L, L st p is infs-preserving holds Image p is continuous LATTICE & Image p
  is infs-inheriting
proof
  let L be continuous complete LATTICE, p be projection Function of L, L such
  that
A1: p is infs-preserving;
  reconsider Lc = {c where c is Element of L: c <= p.c} as non empty Subset of
  L by WAYBEL_1:43;
  reconsider pc = p|Lc as Function of subrelstr Lc, subrelstr Lc by WAYBEL_1:45
;
A2: subrelstr Lc is infs-inheriting by A1,WAYBEL_1:51;
  then
A3: subrelstr Lc is complete by YELLOW_2:30;
A4: pc is infs-preserving
  proof
    let X be Subset of subrelstr Lc;
    assume ex_inf_of X, subrelstr Lc;
    thus ex_inf_of pc.:X, subrelstr Lc by A3,YELLOW_0:17;
    the carrier of subrelstr Lc = Lc by YELLOW_0:def 15;
    then reconsider X9 = X as Subset of L by XBOOLE_1:1;
A5: ex_inf_of X9, L & p preserves_inf_of X9 by A1,YELLOW_0:17;
    X c= the carrier of subrelstr Lc;
    then X c= Lc by YELLOW_0:def 15;
    then
A6: pc.:X = p.:X by RELAT_1:129;
A7: ex_inf_of X, L by YELLOW_0:17;
    then "/\"(X9,L) in the carrier of subrelstr Lc by A2;
    then
A8: dom pc = the carrier of subrelstr Lc & inf X9 = inf X by A7,FUNCT_2:def 1
,YELLOW_0:63;
    ex_inf_of p.:X, L & "/\"(pc.:X,L) in the carrier of subrelstr Lc by A2,
YELLOW_0:17;
    hence inf (pc.:X) = inf (p.:X) by A6,YELLOW_0:63
      .= p.inf X9 by A5
      .= pc.inf X by A8,FUNCT_1:47;
  end;
  reconsider cpc = corestr pc as Function of subrelstr Lc, Image pc;
A9: the carrier of subrelstr rng p = rng p by YELLOW_0:def 15
    .= rng pc by WAYBEL_1:44
    .= the carrier of subrelstr rng pc by YELLOW_0:def 15;
  subrelstr rng pc is full SubRelStr of L by WAYBEL15:1;
  then
A10: Image p = Image pc by A9,YELLOW_0:57;
  pc is closure by WAYBEL_1:47;
  then
A11: cpc is sups-preserving by WAYBEL_1:55;
  subrelstr Lc is sups-inheriting SubRelStr of L by WAYBEL_1:49;
  then subrelstr Lc is continuous LATTICE by A2,WAYBEL_5:28;
  hence Image p is continuous LATTICE by A3,A10,A4,A11,Th19,WAYBEL_5:33;
  thus thesis by A1,A2,WAYBEL_1:51;
end;
