let S, T be non empty with_suprema Poset; :: thesis: for f being Function of S,T st f is directed-sups-preserving holds

f is monotone

let f be Function of S,T; :: thesis: ( f is directed-sups-preserving implies f is monotone )

assume A1: f is directed-sups-preserving ; :: 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} ) )

assume A2: 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} )

A3: y = y "\/" x by A2, YELLOW_0:24;

for a, b being Element of S st a in {x,y} & b in {x,y} holds

ex z being Element of S st

( z in {x,y} & a <= z & b <= z )

then A5: f preserves_sup_of {x,y} by A1;

A6: dom f = the carrier of S by FUNCT_2:def 1;

y <= y ;

then A7: {x,y} is_<=_than y by A2, YELLOW_0:8;

for b being Element of S st {x,y} is_<=_than b holds

y <= b by YELLOW_0:8;

then ex_sup_of {x,y},S by A7, YELLOW_0:30;

then sup (f .: {x,y}) = f . (sup {x,y}) by A5

.= f . y by A3, YELLOW_0:41 ;

then A8: f . y = sup {(f . x),(f . y)} by A6, FUNCT_1:60

.= (f . y) "\/" (f . x) by YELLOW_0:41 ;

let afx, bfy be Element of T; :: thesis: ( not afx = f . x or not bfy = f . y or afx <= bfy )

assume ( afx = f . x & bfy = f . y ) ; :: thesis: afx <= bfy

hence afx <= bfy by A8, YELLOW_0:22; :: thesis: verum

f is monotone

let f be Function of S,T; :: thesis: ( f is directed-sups-preserving implies f is monotone )

assume A1: f is directed-sups-preserving ; :: 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

assume A2: x <= y ; :: thesis: for b

( not b

A3: y = y "\/" x by A2, YELLOW_0:24;

for a, b being Element of S st a in {x,y} & b in {x,y} holds

ex z being Element of S st

( z in {x,y} & a <= z & b <= z )

proof

then
( {x,y} is directed & not {x,y} is empty )
;
let a, b be Element of S; :: thesis: ( a in {x,y} & b in {x,y} implies ex z being Element of S st

( z in {x,y} & a <= z & b <= z ) )

assume A4: ( a in {x,y} & b in {x,y} ) ; :: thesis: ex z being Element of S st

( z in {x,y} & a <= z & b <= z )

take y ; :: thesis: ( y in {x,y} & a <= y & b <= y )

thus y in {x,y} by TARSKI:def 2; :: thesis: ( a <= y & b <= y )

thus ( a <= y & b <= y ) by A2, A4, TARSKI:def 2; :: thesis: verum

end;( z in {x,y} & a <= z & b <= z ) )

assume A4: ( a in {x,y} & b in {x,y} ) ; :: thesis: ex z being Element of S st

( z in {x,y} & a <= z & b <= z )

take y ; :: thesis: ( y in {x,y} & a <= y & b <= y )

thus y in {x,y} by TARSKI:def 2; :: thesis: ( a <= y & b <= y )

thus ( a <= y & b <= y ) by A2, A4, TARSKI:def 2; :: thesis: verum

then A5: f preserves_sup_of {x,y} by A1;

A6: dom f = the carrier of S by FUNCT_2:def 1;

y <= y ;

then A7: {x,y} is_<=_than y by A2, YELLOW_0:8;

for b being Element of S st {x,y} is_<=_than b holds

y <= b by YELLOW_0:8;

then ex_sup_of {x,y},S by A7, YELLOW_0:30;

then sup (f .: {x,y}) = f . (sup {x,y}) by A5

.= f . y by A3, YELLOW_0:41 ;

then A8: f . y = sup {(f . x),(f . y)} by A6, FUNCT_1:60

.= (f . y) "\/" (f . x) by YELLOW_0:41 ;

let afx, bfy be Element of T; :: thesis: ( not afx = f . x or not bfy = f . y or afx <= bfy )

assume ( afx = f . x & bfy = f . y ) ; :: thesis: afx <= bfy

hence afx <= bfy by A8, YELLOW_0:22; :: thesis: verum