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

for a, u being Function of Y,BOOLEAN

for PA being a_partition of Y st u is_independent_of PA,G holds

All ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u

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

for PA being a_partition of Y st u is_independent_of PA,G holds

All ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u

let a, u be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y st u is_independent_of PA,G holds

All ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u

let PA be a_partition of Y; :: thesis: ( u is_independent_of PA,G implies All ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u )

assume A1: u is_independent_of PA,G ; :: thesis: All ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u

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

assume A2: (All ((a 'or' u),PA,G)) . z = TRUE ; :: thesis: ((Ex (a,PA,G)) 'or' u) . z = TRUE

A3: for x being Element of Y holds

( not x in EqClass (z,(CompF (PA,G))) or a . x = TRUE or u . x = TRUE )

for a, u being Function of Y,BOOLEAN

for PA being a_partition of Y st u is_independent_of PA,G holds

All ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u

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

for PA being a_partition of Y st u is_independent_of PA,G holds

All ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u

let a, u be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y st u is_independent_of PA,G holds

All ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u

let PA be a_partition of Y; :: thesis: ( u is_independent_of PA,G implies All ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u )

assume A1: u is_independent_of PA,G ; :: thesis: All ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u

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

assume A2: (All ((a 'or' u),PA,G)) . z = TRUE ; :: thesis: ((Ex (a,PA,G)) 'or' u) . z = TRUE

A3: for x being Element of Y holds

( not x in EqClass (z,(CompF (PA,G))) or a . x = TRUE or u . x = TRUE )

proof

A5:
((Ex (a,PA,G)) 'or' u) . z = ((Ex (a,PA,G)) . z) 'or' (u . z)
by BVFUNC_1:def 4;
let x be Element of Y; :: thesis: ( not x in EqClass (z,(CompF (PA,G))) or a . x = TRUE or u . x = TRUE )

assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: ( a . x = TRUE or u . x = TRUE )

then (a 'or' u) . x = TRUE by A2, BVFUNC_1:def 16;

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

( u . x = TRUE or u . x = FALSE ) by XBOOLEAN:def 3;

hence ( a . x = TRUE or u . x = TRUE ) by A4, BINARITH:3; :: thesis: verum

end;assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: ( a . x = TRUE or u . x = TRUE )

then (a 'or' u) . x = TRUE by A2, BVFUNC_1:def 16;

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

( u . x = TRUE or u . x = FALSE ) by XBOOLEAN:def 3;

hence ( a . x = TRUE or u . x = TRUE ) by A4, BINARITH:3; :: thesis: verum

per cases
( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds

u . x = TRUE or ( ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & not u . x = TRUE ) & ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) ) or ( ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & not u . x = TRUE ) & ( for x being Element of Y holds

( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ) ) ;

end;

u . x = TRUE or ( ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & not u . x = TRUE ) & ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) ) or ( ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & not u . x = TRUE ) & ( for x being Element of Y holds

( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ) ) ;

suppose
for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds

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

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

then ((Ex (a,PA,G)) 'or' u) . z =
((Ex (a,PA,G)) . z) 'or' TRUE
by A5, EQREL_1:def 6

.= TRUE by BINARITH:10 ;

hence ((Ex (a,PA,G)) 'or' u) . z = TRUE ; :: thesis: verum

end;.= TRUE by BINARITH:10 ;

hence ((Ex (a,PA,G)) 'or' u) . z = TRUE ; :: thesis: verum

suppose
( ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & not u . x = TRUE ) & ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) ) ; :: thesis: ((Ex (a,PA,G)) 'or' u) . z = TRUE

( x in EqClass (z,(CompF (PA,G))) & not u . x = TRUE ) & ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) ) ; :: thesis: ((Ex (a,PA,G)) 'or' u) . z = TRUE

then ((Ex (a,PA,G)) 'or' u) . z =
TRUE 'or' (u . z)
by A5, BVFUNC_1:def 17

.= TRUE by BINARITH:10 ;

hence ((Ex (a,PA,G)) 'or' u) . z = TRUE ; :: thesis: verum

end;.= TRUE by BINARITH:10 ;

hence ((Ex (a,PA,G)) 'or' u) . z = TRUE ; :: thesis: verum

suppose A6:
( ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & not u . x = TRUE ) & ( for x being Element of Y holds

( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ) ; :: thesis: ((Ex (a,PA,G)) 'or' u) . z = TRUE

( x in EqClass (z,(CompF (PA,G))) & not u . x = TRUE ) & ( for x being Element of Y holds

( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ) ; :: thesis: ((Ex (a,PA,G)) 'or' u) . z = TRUE

A7:
z in EqClass (z,(CompF (PA,G)))
by EQREL_1:def 6;

A8: a . z <> TRUE by A6, EQREL_1:def 6;

consider x1 being Element of Y such that

A9: x1 in EqClass (z,(CompF (PA,G))) and

A10: u . x1 <> TRUE by A6;

u . x1 = u . z by A1, A7, A9, BVFUNC_1:def 15;

hence ((Ex (a,PA,G)) 'or' u) . z = TRUE by A3, A8, A10, EQREL_1:def 6; :: thesis: verum

end;A8: a . z <> TRUE by A6, EQREL_1:def 6;

consider x1 being Element of Y such that

A9: x1 in EqClass (z,(CompF (PA,G))) and

A10: u . x1 <> TRUE by A6;

u . x1 = u . z by A1, A7, A9, BVFUNC_1:def 15;

hence ((Ex (a,PA,G)) 'or' u) . z = TRUE by A3, A8, A10, EQREL_1:def 6; :: thesis: verum