
theorem Th11:

:: 1.2. LEMMA (ii), p. 143
  for S,T being lower complete TopLattice for f being Function of
  S,T st f is continuous holds f is filtered-infs-preserving
proof
  let S,T be lower complete TopLattice;
  reconsider BB = the set of all (uparrow x)` where x is Element of S as
  prebasis of S by Def1;
  let f be Function of S,T such that
A1: f is continuous;
  let F be Subset of S such that
A2: F is non empty filtered and
  ex_inf_of F,S;
  for A being Subset of S st A in BB & inf F in A holds F meets A
  proof
    let A be Subset of S;
    assume A in BB;
    then consider x being Element of S such that
A3: A = (uparrow x)`;
    assume inf F in A;
    then not inf F >= x by A3,YELLOW_9:1;
    then not F is_>=_than x by YELLOW_0:33;
    then consider y being Element of S such that
A4: y in F and
A5: not y >= x;
    y in A by A3,A5,YELLOW_9:1;
    hence thesis by A4,XBOOLE_0:3;
  end;
  then
A6: inf F in Cl F by A2,Th10;
A7: f is monotone
  proof
    let x,y be Element of S such that
A8: x <= y;
    f.x <= f.x;
    then f.x in uparrow (f.x) by WAYBEL_0:18;
    then
A9: x in f"uparrow (f.x) by FUNCT_2:38;
    uparrow (f.x) is closed by Th4;
    then f"uparrow (f.x) is closed by A1;
    then f"uparrow (f.x) is upper by Th6;
    then y in f"uparrow (f.x) by A9,A8;
    then f.y in uparrow (f.x) by FUNCT_2:38;
    hence thesis by WAYBEL_0:18;
  end;
  f.inf F is_<=_than f.:F
  proof
    let x be Element of T;
    assume x in f.:F;
    then consider a being object such that
A10: a in the carrier of S and
A11: a in F and
A12: x = f.a by FUNCT_2:64;
    reconsider a as Element of S by A10;
    inf F is_<=_than F by YELLOW_0:33;
    then inf F <= a by A11;
    hence thesis by A7,A12;
  end;
  then
A13: f.inf F <= inf (f.:F) by YELLOW_0:33;
  thus ex_inf_of f.:F,T by YELLOW_0:17;
  F c= f"uparrow inf (f.:F)
  proof
    let x be object;
    assume
A14: x in F;
    then reconsider s = x as Element of S;
    f.s in f.:F by A14,FUNCT_2:35;
    then inf (f.:F) <= f.s by YELLOW_2:22;
    then f.s in uparrow inf (f.:F) by WAYBEL_0:18;
    hence thesis by FUNCT_2:38;
  end;
  then
A15: Cl F c= Cl (f"uparrow inf (f.:F)) by PRE_TOPC:19;
  uparrow inf (f.:F) is closed by Th4;
  then f"uparrow inf (f.:F) is closed by A1;
  then Cl F c= f"uparrow inf (f.:F) by A15,PRE_TOPC:22;
  then f.inf F in uparrow inf (f.:F) by A6,FUNCT_2:38;
  then f.inf F >= inf (f.:F) by WAYBEL_0:18;
  hence inf (f.:F) = f.inf F by A13,ORDERS_2:2;
end;
