let Y be non empty set ; for G being Subset of (PARTITIONS Y)
for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds (Ex (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' (All (a,PA,G)) 'imp' (Ex (b,PA,G))
let G be Subset of (PARTITIONS Y); for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds (Ex (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' (All (a,PA,G)) 'imp' (Ex (b,PA,G))
let a, b be Function of Y,BOOLEAN; for PA being a_partition of Y holds (Ex (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' (All (a,PA,G)) 'imp' (Ex (b,PA,G))
let PA be a_partition of Y; (Ex (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' (All (a,PA,G)) 'imp' (Ex (b,PA,G))
A1:
Ex (a,PA,G) = B_SUP (a,(CompF (PA,G)))
by BVFUNC_2:def 10;
let z be Element of Y; BVFUNC_1:def 12 ( not ((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE or ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE )
A2:
( 'not' ((Ex (a,PA,G)) . z) = TRUE or 'not' ((Ex (a,PA,G)) . z) = FALSE )
by XBOOLEAN:def 3;
assume
((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE
; ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE
then A3:
('not' ((Ex (a,PA,G)) . z)) 'or' ((Ex (b,PA,G)) . z) = TRUE
by BVFUNC_1:def 8;
A4:
z in EqClass (z,(CompF (PA,G)))
by EQREL_1:def 6;
per cases
( 'not' ((Ex (a,PA,G)) . z) = TRUE or (Ex (b,PA,G)) . z = TRUE )
by A3, A2, BINARITH:3;
suppose
'not' ((Ex (a,PA,G)) . z) = TRUE
;
((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE then
(Ex (a,PA,G)) . z = FALSE
by MARGREL1:11;
then
a . z <> TRUE
by A1, A4, BVFUNC_1:def 17;
then
(B_INF (a,(CompF (PA,G)))) . z = FALSE
by A4, BVFUNC_1:def 16;
then
(All (a,PA,G)) . z = FALSE
by BVFUNC_2:def 9;
hence ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z =
('not' FALSE) 'or' ((Ex (b,PA,G)) . z)
by BVFUNC_1:def 8
.=
TRUE 'or' ((Ex (b,PA,G)) . z)
by MARGREL1:11
.=
TRUE
by BINARITH:10
;
verum end; end;