let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a, b being Element of Funcs 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); :: thesis: for a, b being Element of Funcs 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 Element of Funcs Y,BOOLEAN ; :: thesis: 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; :: thesis: (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; :: according to BVFUNC_1:def 15 :: thesis: ( not ((Ex a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE or ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE )
assume
((Ex a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE
; :: thesis: ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE
then A2:
('not' ((Ex a,PA,G) . z)) 'or' ((Ex b,PA,G) . z) = TRUE
by BVFUNC_1:def 11;
A3:
( 'not' ((Ex a,PA,G) . z) = TRUE or 'not' ((Ex a,PA,G) . z) = FALSE )
by XBOOLEAN:def 3;
A4:
( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G )
by EQREL_1:def 8;
per cases
( 'not' ((Ex a,PA,G) . z) = TRUE or (Ex b,PA,G) . z = TRUE )
by A2, A3, BINARITH:7;
suppose
'not' ((Ex a,PA,G) . z) = TRUE
;
:: thesis: ((All a,PA,G) 'imp' (Ex b,PA,G)) . z = TRUE then
(Ex a,PA,G) . z = FALSE
by MARGREL1:41;
then
a . z <> TRUE
by A1, A4, BVFUNC_1:def 20;
then
(B_INF a,(CompF PA,G)) . z = FALSE
by A4, BVFUNC_1:def 19;
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 11
.=
TRUE 'or' ((Ex b,PA,G) . z)
by MARGREL1:41
.=
TRUE
by BINARITH:19
;
:: thesis: verum end;