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 b_{1}, b_{2} being Element of the carrier of T holds

( not b_{1} = f . x or not b_{2} = f . y or b_{1} <= b_{2} ) )

A2: ex_inf_of {x},S by YELLOW_0:38;

A3: ex_inf_of {y},S by YELLOW_0:38;

A4: f preserves_inf_of uparrow x by A1;

A5: f preserves_inf_of uparrow y by A1;

A6: ex_inf_of uparrow x,S by A2, Th37;

A7: ex_inf_of uparrow y,S by A3, Th37;

A8: ex_inf_of f .: (uparrow x),T by A4, A6;

A9: ex_inf_of f .: (uparrow y),T by A5, A7;

A10: inf (f .: (uparrow x)) = f . (inf (uparrow x)) by A4, A6;

A11: inf (f .: (uparrow y)) = f . (inf (uparrow y)) by A5, A7;

assume x <= y ; :: thesis: for b_{1}, b_{2} being Element of the carrier of T holds

( not b_{1} = f . x or not b_{2} = f . y or b_{1} <= b_{2} )

then A12: uparrow y c= uparrow x by Th22;

A13: inf (f .: (uparrow x)) = f . (inf {x}) by A10, Th38, YELLOW_0:38;

A14: inf (f .: (uparrow y)) = f . (inf {y}) by A11, Th38, YELLOW_0:38;

A15: inf (f .: (uparrow x)) = f . x by A13, YELLOW_0:39;

inf (f .: (uparrow y)) = f . y by A14, YELLOW_0:39;

hence for b_{1}, b_{2} being Element of the carrier of T holds

( not b_{1} = f . x or not b_{2} = f . y or b_{1} <= b_{2} )
by A8, A9, A12, A15, RELAT_1:123, YELLOW_0:35; :: thesis: verum

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 b

( not b

A2: ex_inf_of {x},S by YELLOW_0:38;

A3: ex_inf_of {y},S by YELLOW_0:38;

A4: f preserves_inf_of uparrow x by A1;

A5: f preserves_inf_of uparrow y by A1;

A6: ex_inf_of uparrow x,S by A2, Th37;

A7: ex_inf_of uparrow y,S by A3, Th37;

A8: ex_inf_of f .: (uparrow x),T by A4, A6;

A9: ex_inf_of f .: (uparrow y),T by A5, A7;

A10: inf (f .: (uparrow x)) = f . (inf (uparrow x)) by A4, A6;

A11: inf (f .: (uparrow y)) = f . (inf (uparrow y)) by A5, A7;

assume x <= y ; :: thesis: for b

( not b

then A12: uparrow y c= uparrow x by Th22;

A13: inf (f .: (uparrow x)) = f . (inf {x}) by A10, Th38, YELLOW_0:38;

A14: inf (f .: (uparrow y)) = f . (inf {y}) by A11, Th38, YELLOW_0:38;

A15: inf (f .: (uparrow x)) = f . x by A13, YELLOW_0:39;

inf (f .: (uparrow y)) = f . y by A14, YELLOW_0:39;

hence for b

( not b