let S, T be non empty Poset; :: thesis: for f being Function of S,T st ( for X being Filter of S holds f preserves_inf_of X ) holds
f is monotone
let f be Function of S,T; :: thesis: ( ( for X being Filter of S holds f preserves_inf_of X ) implies f is monotone )
assume A1: for X being Filter of S holds f preserves_inf_of X ; :: thesis: f is monotone
let x, y be Element of S; :: according to ORDERS_3:def 5 :: thesis: ( not x <= y or for b₁, b₂ being Element of the carrier of T holds
( not b₁ = f . x or not b₂ = f . y or b₁ <= b₂ ) )
A2: ( ex_inf_of {x},S & ex_inf_of {y},S ) by YELLOW_0:38;
then ( f preserves_inf_of uparrow x & f preserves_inf_of uparrow y & ex_inf_of uparrow x,S & ex_inf_of uparrow y,S ) by A1, Th37;
then A3: ( ex_inf_of f .: (uparrow x),T & ex_inf_of f .: (uparrow y),T & inf (f .: (uparrow x)) = f . (inf (uparrow x)) & inf (f .: (uparrow y)) = f . (inf (uparrow y)) ) by Def30;
assume x <= y ; :: thesis: for b₁, b₂ being Element of the carrier of T holds
( not b₁ = f . x or not b₂ = f . y or b₁ <= b₂ )
then A4: uparrow y c= uparrow x by Th22;
( inf (f .: (uparrow x)) = f . (inf {x}) & inf (f .: (uparrow y)) = f . (inf {y}) ) by A2, A3, Th38;
then ( inf (f .: (uparrow x)) = f . x & inf (f .: (uparrow y)) = f . y ) by YELLOW_0:39;
hence for b₁, b₂ being Element of the carrier of T holds
( not b₁ = f . x or not b₂ = f . y or b₁ <= b₂ ) by A3, A4, RELAT_1:156, YELLOW_0:35; :: thesis: verum