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 a 'or' b '<' (Ex a,PA,G) 'or' (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 a 'or' b '<' (Ex a,PA,G) 'or' (Ex b,PA,G)
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds a 'or' b '<' (Ex a,PA,G) 'or' (Ex b,PA,G)
let PA be a_partition of Y; :: thesis: a 'or' b '<' (Ex a,PA,G) 'or' (Ex b,PA,G)
A1:
Ex a,PA,G = B_SUP a,(CompF PA,G)
by BVFUNC_2:def 10;
A2:
Ex b,PA,G = B_SUP b,(CompF PA,G)
by BVFUNC_2:def 10;
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (a 'or' b) . z = TRUE or ((Ex a,PA,G) 'or' (Ex b,PA,G)) . z = TRUE )
assume
(a 'or' b) . z = TRUE
; :: thesis: ((Ex a,PA,G) 'or' (Ex b,PA,G)) . z = TRUE
then A3:
(a . z) 'or' (b . z) = TRUE
by BVFUNC_1:def 7;
A4:
( b . z = TRUE or b . z = FALSE )
by XBOOLEAN:def 3;
A5:
( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G )
by EQREL_1:def 8;
per cases
( a . z = TRUE or b . z = TRUE )
by A3, A4, BINARITH:7;
suppose A6:
a . z = TRUE
;
:: thesis: ((Ex a,PA,G) 'or' (Ex b,PA,G)) . z = TRUE thus ((Ex a,PA,G) 'or' (Ex b,PA,G)) . z =
((Ex a,PA,G) . z) 'or' ((Ex b,PA,G) . z)
by BVFUNC_1:def 7
.=
TRUE 'or' ((Ex b,PA,G) . z)
by A1, A5, A6, BVFUNC_1:def 20
.=
TRUE
by BINARITH:19
;
:: thesis: verum end; suppose A7:
b . z = TRUE
;
:: thesis: ((Ex a,PA,G) 'or' (Ex b,PA,G)) . z = TRUE thus ((Ex a,PA,G) 'or' (Ex b,PA,G)) . z =
((Ex a,PA,G) . z) 'or' ((Ex b,PA,G) . z)
by BVFUNC_1:def 7
.=
((Ex a,PA,G) . z) 'or' TRUE
by A2, A5, A7, BVFUNC_1:def 20
.=
TRUE
by BINARITH:19
;
:: thesis: verum end;