let H be distributive complete LATTICE; 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; for X being finite Subset of H holds inf ({a} "\/" X) = a "\/" (inf X)
let X be finite Subset of H; inf ({a} "\/" X) = a "\/" (inf X)
defpred S1[ set ] means ex A being Subset of H st
( A = $1 & a "\/" (inf A) = inf ({a} "\/" A) );
A1:
S1[ {} ]
A2:
for x, B being set st x in X & B c= X & S1[B] holds
S1[B \/ {x}]
proof
let x,
B be
set ;
( x in X & B c= X & S1[B] implies S1[B \/ {x}] )
assume that A3:
x in X
and A4:
B c= X
and A5:
S1[
B]
;
S1[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
;
( C = B \/ {x} & a "\/" (inf C) = inf ({a} "\/" C) )
thus
C = B \/ {x}
;
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
;
verum
end;
A11:
X is finite
;
S1[X]
from FINSET_1:sch 2(A11, A1, A2);
hence
inf ({a} "\/" X) = a "\/" (inf X)
; verum