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 (All (a,PA,G)) 'imp' (Ex (b,PA,G)) = Ex ((a 'imp' 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 (All (a,PA,G)) 'imp' (Ex (b,PA,G)) = Ex ((a 'imp' b),PA,G)
let a, b be Function of Y,BOOLEAN; for PA being a_partition of Y holds (All (a,PA,G)) 'imp' (Ex (b,PA,G)) = Ex ((a 'imp' b),PA,G)
let PA be a_partition of Y; (All (a,PA,G)) 'imp' (Ex (b,PA,G)) = Ex ((a 'imp' b),PA,G)
A1:
All (a,PA,G) = B_INF (a,(CompF (PA,G)))
by BVFUNC_2:def 9;
A2:
(All (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' Ex ((a 'imp' b),PA,G)
proof
let z be
Element of
Y;
BVFUNC_1:def 12 ( not ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE or (Ex ((a 'imp' b),PA,G)) . z = TRUE )
A3:
(
'not' ((All (a,PA,G)) . z) = TRUE or
'not' ((All (a,PA,G)) . z) = FALSE )
by XBOOLEAN:def 3;
assume
((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE
;
(Ex ((a 'imp' b),PA,G)) . z = TRUE
then A4:
('not' ((All (a,PA,G)) . z)) 'or' ((Ex (b,PA,G)) . z) = TRUE
by BVFUNC_1:def 8;
per cases
( 'not' ((All (a,PA,G)) . z) = TRUE or (Ex (b,PA,G)) . z = TRUE )
by A4, A3, BINARITH:3;
suppose
'not' ((All (a,PA,G)) . z) = TRUE
;
(Ex ((a 'imp' b),PA,G)) . z = TRUE then
(All (a,PA,G)) . z = FALSE
by MARGREL1:11;
then consider x1 being
Element of
Y such that A5:
x1 in EqClass (
z,
(CompF (PA,G)))
and A6:
a . x1 <> TRUE
by A1, BVFUNC_1:def 16;
(a 'imp' b) . x1 =
('not' (a . x1)) 'or' (b . x1)
by BVFUNC_1:def 8
.=
('not' FALSE) 'or' (b . x1)
by A6, XBOOLEAN:def 3
.=
TRUE 'or' (b . x1)
by MARGREL1:11
.=
TRUE
by BINARITH:10
;
then
(B_SUP ((a 'imp' b),(CompF (PA,G)))) . z = TRUE
by A5, BVFUNC_1:def 17;
hence
(Ex ((a 'imp' b),PA,G)) . z = TRUE
by BVFUNC_2:def 10;
verum end; end;
end;
Ex ((a 'imp' b),PA,G) '<' (All (a,PA,G)) 'imp' (Ex (b,PA,G))
proof
let z be
Element of
Y;
BVFUNC_1:def 12 ( not (Ex ((a 'imp' b),PA,G)) . z = TRUE or ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE )
assume A10:
(Ex ((a 'imp' b),PA,G)) . z = TRUE
;
((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE
then consider x1 being
Element of
Y such that A11:
x1 in EqClass (
z,
(CompF (PA,G)))
and A12:
(a 'imp' b) . x1 = TRUE
;
A13:
('not' (a . x1)) 'or' (b . x1) = TRUE
by A12, BVFUNC_1:def 8;
A14:
(
'not' (a . x1) = TRUE or
'not' (a . x1) = FALSE )
by XBOOLEAN:def 3;
per cases
( 'not' (a . x1) = TRUE or b . x1 = TRUE )
by A13, A14, BINARITH:3;
suppose
'not' (a . x1) = TRUE
;
((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE then
a . x1 = FALSE
by MARGREL1:11;
then
(B_INF (a,(CompF (PA,G)))) . z = FALSE
by A11, BVFUNC_1:def 16;
hence ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z =
('not' FALSE) 'or' ((Ex (b,PA,G)) . z)
by A1, BVFUNC_1:def 8
.=
TRUE 'or' ((Ex (b,PA,G)) . z)
by MARGREL1:11
.=
TRUE
by BINARITH:10
;
verum end; suppose
b . x1 = TRUE
;
((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE then
(B_SUP (b,(CompF (PA,G)))) . z = TRUE
by A11, BVFUNC_1:def 17;
then
(Ex (b,PA,G)) . z = TRUE
by BVFUNC_2:def 10;
hence ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z =
('not' ((All (a,PA,G)) . z)) 'or' TRUE
by BVFUNC_1:def 8
.=
TRUE
by BINARITH:10
;
verum end; end;
end;
hence
(All (a,PA,G)) 'imp' (Ex (b,PA,G)) = Ex ((a 'imp' b),PA,G)
by A2, BVFUNC_1:15; verum