let H be distributive complete LATTICE; :: thesis: for a being Element of H

for X being finite Subset of H holds inf ({a} "\/" X) = a "\/" (inf X)

let a be Element of H; :: thesis: for X being finite Subset of H holds inf ({a} "\/" X) = a "\/" (inf X)

let X be finite Subset of H; :: thesis: inf ({a} "\/" X) = a "\/" (inf X)

defpred S_{1}[ set ] means ex A being Subset of H st

( A = $1 & a "\/" (inf A) = inf ({a} "\/" A) );

A1: S_{1}[ {} ]
_{1}[B] holds

S_{1}[B \/ {x}]

S_{1}[X]
from FINSET_1:sch 2(A11, A1, A2);

hence inf ({a} "\/" X) = a "\/" (inf X) ; :: thesis: verum

for X being finite Subset of H holds inf ({a} "\/" X) = a "\/" (inf X)

let a be Element of H; :: thesis: for X being finite Subset of H holds inf ({a} "\/" X) = a "\/" (inf X)

let X be finite Subset of H; :: thesis: inf ({a} "\/" X) = a "\/" (inf X)

defpred S

( A = $1 & a "\/" (inf A) = inf ({a} "\/" A) );

A1: S

proof

A2:
for x, B being set st x in X & B c= X & S
reconsider A = {} as Subset of H by XBOOLE_1:2;

take A ; :: thesis: ( A = {} & a "\/" (inf A) = inf ({a} "\/" A) )

thus A = {} ; :: thesis: a "\/" (inf A) = inf ({a} "\/" A)

( a <= Top H & {a} "\/" ({} H) = {} ) by YELLOW_0:45, YELLOW_4:9;

hence a "\/" (inf A) = inf ({a} "\/" A) by YELLOW_0:24; :: thesis: verum

end;take A ; :: thesis: ( A = {} & a "\/" (inf A) = inf ({a} "\/" A) )

thus A = {} ; :: thesis: a "\/" (inf A) = inf ({a} "\/" A)

( a <= Top H & {a} "\/" ({} H) = {} ) by YELLOW_0:45, YELLOW_4:9;

hence a "\/" (inf A) = inf ({a} "\/" A) by YELLOW_0:24; :: thesis: verum

S

proof

A11:
X is finite
;
let x, B be set ; :: thesis: ( x in X & B c= X & S_{1}[B] implies S_{1}[B \/ {x}] )

assume that

A3: x in X and

A4: B c= X and

A5: S_{1}[B]
; :: thesis: S_{1}[B \/ {x}]

reconsider x1 = x as Element of H by A3;

A6: {x1} c= the carrier of H ;

B c= the carrier of H by A4, XBOOLE_1:1;

then reconsider C = B \/ {x} as Subset of H by A6, XBOOLE_1:8;

take C ; :: thesis: ( C = B \/ {x} & a "\/" (inf C) = inf ({a} "\/" C) )

thus C = B \/ {x} ; :: thesis: a "\/" (inf C) = inf ({a} "\/" C)

consider A being Subset of H such that

A7: A = B and

A8: a "\/" (inf A) = inf ({a} "\/" A) by A5;

A9: {a} "\/" C = ({a} "\/" A) \/ ({a} "\/" {x1}) by A7, YELLOW_4:16

.= ({a} "\/" A) \/ {(a "\/" x1)} by YELLOW_4:19 ;

A10: ( ex_inf_of {a} "\/" A,H & ex_inf_of {(a "\/" x1)},H ) by YELLOW_0:17;

( ex_inf_of B,H & ex_inf_of {x},H ) by YELLOW_0:17;

hence a "\/" (inf C) = a "\/" (("/\" (B,H)) "/\" ("/\" ({x},H))) by YELLOW_2:4

.= (inf ({a} "\/" A)) "/\" (a "\/" ("/\" ({x},H))) by A7, A8, WAYBEL_1:5

.= (inf ({a} "\/" A)) "/\" (a "\/" x1) by YELLOW_0:39

.= (inf ({a} "\/" A)) "/\" (inf {(a "\/" x1)}) by YELLOW_0:39

.= inf ({a} "\/" C) by A10, A9, YELLOW_2:4 ;

:: thesis: verum

end;assume that

A3: x in X and

A4: B c= X and

A5: S

reconsider x1 = x as Element of H by A3;

A6: {x1} c= the carrier of H ;

B c= the carrier of H by A4, XBOOLE_1:1;

then reconsider C = B \/ {x} as Subset of H by A6, XBOOLE_1:8;

take C ; :: thesis: ( C = B \/ {x} & a "\/" (inf C) = inf ({a} "\/" C) )

thus C = B \/ {x} ; :: thesis: a "\/" (inf C) = inf ({a} "\/" C)

consider A being Subset of H such that

A7: A = B and

A8: a "\/" (inf A) = inf ({a} "\/" A) by A5;

A9: {a} "\/" C = ({a} "\/" A) \/ ({a} "\/" {x1}) by A7, YELLOW_4:16

.= ({a} "\/" A) \/ {(a "\/" x1)} by YELLOW_4:19 ;

A10: ( ex_inf_of {a} "\/" A,H & ex_inf_of {(a "\/" x1)},H ) by YELLOW_0:17;

( ex_inf_of B,H & ex_inf_of {x},H ) by YELLOW_0:17;

hence a "\/" (inf C) = a "\/" (("/\" (B,H)) "/\" ("/\" ({x},H))) by YELLOW_2:4

.= (inf ({a} "\/" A)) "/\" (a "\/" ("/\" ({x},H))) by A7, A8, WAYBEL_1:5

.= (inf ({a} "\/" A)) "/\" (a "\/" x1) by YELLOW_0:39

.= (inf ({a} "\/" A)) "/\" (inf {(a "\/" x1)}) by YELLOW_0:39

.= inf ({a} "\/" C) by A10, A9, YELLOW_2:4 ;

:: thesis: verum

S

hence inf ({a} "\/" X) = a "\/" (inf X) ; :: thesis: verum