let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)

for a, b being Function of Y,BOOLEAN

for PA being a_partition of Y holds a 'imp' b '<' a 'imp' (Ex (b,PA,G))

let G be Subset of (PARTITIONS Y); :: thesis: for a, b being Function of Y,BOOLEAN

for PA being a_partition of Y holds a 'imp' b '<' a 'imp' (Ex (b,PA,G))

let a, b be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds a 'imp' b '<' a 'imp' (Ex (b,PA,G))

let PA be a_partition of Y; :: thesis: a 'imp' b '<' a 'imp' (Ex (b,PA,G))

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (a 'imp' b) . z = TRUE or (a 'imp' (Ex (b,PA,G))) . z = TRUE )

A1: ( 'not' (a . z) = TRUE or 'not' (a . z) = FALSE ) by XBOOLEAN:def 3;

A2: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;

assume (a 'imp' b) . z = TRUE ; :: thesis: (a 'imp' (Ex (b,PA,G))) . z = TRUE

then A3: ('not' (a . z)) 'or' (b . z) = TRUE by BVFUNC_1:def 8;

for a, b being Function of Y,BOOLEAN

for PA being a_partition of Y holds a 'imp' b '<' a 'imp' (Ex (b,PA,G))

let G be Subset of (PARTITIONS Y); :: thesis: for a, b being Function of Y,BOOLEAN

for PA being a_partition of Y holds a 'imp' b '<' a 'imp' (Ex (b,PA,G))

let a, b be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds a 'imp' b '<' a 'imp' (Ex (b,PA,G))

let PA be a_partition of Y; :: thesis: a 'imp' b '<' a 'imp' (Ex (b,PA,G))

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (a 'imp' b) . z = TRUE or (a 'imp' (Ex (b,PA,G))) . z = TRUE )

A1: ( 'not' (a . z) = TRUE or 'not' (a . z) = FALSE ) by XBOOLEAN:def 3;

A2: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;

assume (a 'imp' b) . z = TRUE ; :: thesis: (a 'imp' (Ex (b,PA,G))) . z = TRUE

then A3: ('not' (a . z)) 'or' (b . z) = TRUE by BVFUNC_1:def 8;

per cases
( 'not' (a . z) = TRUE or b . z = TRUE )
by A3, A1, BINARITH:3;

end;

suppose
'not' (a . z) = TRUE
; :: thesis: (a 'imp' (Ex (b,PA,G))) . z = TRUE

hence (a 'imp' (Ex (b,PA,G))) . z =
TRUE 'or' ((Ex (b,PA,G)) . z)
by BVFUNC_1:def 8

.= TRUE by BINARITH:10 ;

:: thesis: verum

end;.= TRUE by BINARITH:10 ;

:: thesis: verum

suppose
b . z = TRUE
; :: thesis: (a 'imp' (Ex (b,PA,G))) . z = TRUE

then
(B_SUP (b,(CompF (PA,G)))) . z = TRUE
by A2, BVFUNC_1:def 17;

then (Ex (b,PA,G)) . z = TRUE by BVFUNC_2:def 10;

hence (a 'imp' (Ex (b,PA,G))) . z = ('not' (a . z)) 'or' TRUE by BVFUNC_1:def 8

.= TRUE by BINARITH:10 ;

:: thesis: verum

end;then (Ex (b,PA,G)) . z = TRUE by BVFUNC_2:def 10;

hence (a 'imp' (Ex (b,PA,G))) . z = ('not' (a . z)) 'or' TRUE by BVFUNC_1:def 8

.= TRUE by BINARITH:10 ;

:: thesis: verum