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 'or' b '<' (Ex (a,PA,G)) 'or' (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 'or' b '<' (Ex (a,PA,G)) 'or' (Ex (b,PA,G))

let a, b be Function of 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;

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

A2: Ex (b,PA,G) = B_SUP (b,(CompF (PA,G))) by BVFUNC_2:def 10;

A3: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;

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

then A4: (a . z) 'or' (b . z) = TRUE by BVFUNC_1:def 4;

A5: ( b . z = TRUE or b . z = FALSE ) by XBOOLEAN:def 3;

for a, b being Function of 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 Function of 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 Function of 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;

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

A2: Ex (b,PA,G) = B_SUP (b,(CompF (PA,G))) by BVFUNC_2:def 10;

A3: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;

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

then A4: (a . z) 'or' (b . z) = TRUE by BVFUNC_1:def 4;

A5: ( b . z = TRUE or b . z = FALSE ) by XBOOLEAN:def 3;

per cases
( a . z = TRUE or b . z = TRUE )
by A4, A5, BINARITH:3;

end;

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 4

.= TRUE 'or' ((Ex (b,PA,G)) . z) by A1, A3, A6, BVFUNC_1:def 17

.= TRUE by BINARITH:10 ; :: thesis: verum

end;.= TRUE 'or' ((Ex (b,PA,G)) . z) by A1, A3, A6, BVFUNC_1:def 17

.= TRUE by BINARITH:10 ; :: thesis: verum

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 4

.= ((Ex (a,PA,G)) . z) 'or' TRUE by A2, A3, A7, BVFUNC_1:def 17

.= TRUE by BINARITH:10 ; :: thesis: verum

end;.= ((Ex (a,PA,G)) . z) 'or' TRUE by A2, A3, A7, BVFUNC_1:def 17

.= TRUE by BINARITH:10 ; :: thesis: verum