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 'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) = (All (('not' a),PA,G)) 'or' (All (('not' 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 'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) = (All (('not' a),PA,G)) 'or' (All (('not' b),PA,G))
let a, b be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds 'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) = (All (('not' a),PA,G)) 'or' (All (('not' b),PA,G))
let PA be a_partition of Y; :: thesis: 'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) = (All (('not' a),PA,G)) 'or' (All (('not' b),PA,G))
A1: All (('not' b),PA,G) = B_INF (('not' b),(CompF (PA,G))) by BVFUNC_2:def 9;
A2: Ex (b,PA,G) = B_SUP (b,(CompF (PA,G))) by BVFUNC_2:def 10;
A3: Ex (a,PA,G) = B_SUP (a,(CompF (PA,G))) by BVFUNC_2:def 10;
A4: (All (('not' a),PA,G)) 'or' (All (('not' b),PA,G)) '<' 'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))

proof

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ((All (('not' a),PA,G)) 'or' (All (('not' b),PA,G))) . z = TRUE or ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = TRUE )
A5: ( (All (('not' b),PA,G)) . z = TRUE or (All (('not' b),PA,G)) . z = FALSE ) by XBOOLEAN:def 3;
assume ((All (('not' a),PA,G)) 'or' (All (('not' b),PA,G))) . z = TRUE ; :: thesis: ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = TRUE
then A6: ((All (('not' a),PA,G)) . z) 'or' ((All (('not' b),PA,G)) . z) = TRUE by BVFUNC_1:def 4;

per cases ( (All (('not' a),PA,G)) . z = TRUE or (All (('not' b),PA,G)) . z = TRUE ) by A6, A5, BINARITH:3;

suppose A7: (All (('not' a),PA,G)) . z = TRUE ; :: thesis: ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = TRUE

A8: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
('not' a) . x = TRUE

assume ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not ('not' a) . x = TRUE ) ; :: thesis: contradiction
then (B_INF (('not' a),(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 16;
hence contradiction by A7, BVFUNC_2:def 9; :: thesis: verum

end;

A9: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
a . x <> TRUE

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies a . x <> TRUE )
assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: a . x <> TRUE
then ('not' a) . x = TRUE by A8;
then 'not' (a . x) = TRUE by MARGREL1:def 19;
hence a . x <> TRUE by MARGREL1:11; :: thesis: verum

end;

thus ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = 'not' (((Ex (a,PA,G)) '&' (Ex (b,PA,G))) . z) by MARGREL1:def 19
.= 'not' (((Ex (a,PA,G)) . z) '&' ((Ex (b,PA,G)) . z)) by MARGREL1:def 20
.= 'not' (FALSE '&' ((Ex (b,PA,G)) . z)) by A3, A9, BVFUNC_1:def 17
.= 'not' FALSE by MARGREL1:12
.= TRUE by MARGREL1:11 ; :: thesis: verum

end;

suppose A10: (All (('not' b),PA,G)) . z = TRUE ; :: thesis: ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = TRUE

A11: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
('not' b) . x = TRUE

assume ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not ('not' b) . x = TRUE ) ; :: thesis: contradiction
then (B_INF (('not' b),(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 16;
hence contradiction by A10, BVFUNC_2:def 9; :: thesis: verum

end;

A12: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
b . x <> TRUE

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies b . x <> TRUE )
assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: b . x <> TRUE
then ('not' b) . x = TRUE by A11;
then 'not' (b . x) = TRUE by MARGREL1:def 19;
hence b . x <> TRUE by MARGREL1:11; :: thesis: verum

end;

thus ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = 'not' (((Ex (a,PA,G)) '&' (Ex (b,PA,G))) . z) by MARGREL1:def 19
.= 'not' (((Ex (a,PA,G)) . z) '&' ((Ex (b,PA,G)) . z)) by MARGREL1:def 20
.= 'not' (((Ex (a,PA,G)) . z) '&' FALSE) by A2, A12, BVFUNC_1:def 17
.= 'not' FALSE by MARGREL1:12
.= TRUE by MARGREL1:11 ; :: thesis: verum

end;

end;

end;

A13: All (('not' a),PA,G) = B_INF (('not' a),(CompF (PA,G))) by BVFUNC_2:def 9;
'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) '<' (All (('not' a),PA,G)) 'or' (All (('not' b),PA,G))

proof

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = TRUE or ((All (('not' a),PA,G)) 'or' (All (('not' b),PA,G))) . z = TRUE )
assume ('not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G)))) . z = TRUE ; :: thesis: ((All (('not' a),PA,G)) 'or' (All (('not' b),PA,G))) . z = TRUE
then 'not' (((Ex (a,PA,G)) '&' (Ex (b,PA,G))) . z) = TRUE by MARGREL1:def 19;
then ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) . z = FALSE by MARGREL1:11;
then A14: ((Ex (a,PA,G)) . z) '&' ((Ex (b,PA,G)) . z) = FALSE by MARGREL1:def 20;

per cases ( (Ex (a,PA,G)) . z = FALSE or (Ex (b,PA,G)) . z = FALSE ) by A14, MARGREL1:12;

suppose A15: (Ex (a,PA,G)) . z = FALSE ; :: thesis: ((All (('not' a),PA,G)) 'or' (All (('not' b),PA,G))) . z = TRUE

A16: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
('not' a) . x = TRUE

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies ('not' a) . x = TRUE )
assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: ('not' a) . x = TRUE
then a . x <> TRUE by A3, A15, BVFUNC_1:def 17;
then a . x = FALSE by XBOOLEAN:def 3;
then 'not' (a . x) = TRUE by MARGREL1:11;
hence ('not' a) . x = TRUE by MARGREL1:def 19; :: thesis: verum

end;

thus ((All (('not' a),PA,G)) 'or' (All (('not' b),PA,G))) . z = ((All (('not' a),PA,G)) . z) 'or' ((All (('not' b),PA,G)) . z) by BVFUNC_1:def 4
.= TRUE 'or' ((All (('not' b),PA,G)) . z) by A13, A16, BVFUNC_1:def 16
.= TRUE by BINARITH:10 ; :: thesis: verum

end;

suppose A17: (Ex (b,PA,G)) . z = FALSE ; :: thesis: ((All (('not' a),PA,G)) 'or' (All (('not' b),PA,G))) . z = TRUE

A18: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
('not' b) . x = TRUE

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies ('not' b) . x = TRUE )
assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: ('not' b) . x = TRUE
then b . x <> TRUE by A2, A17, BVFUNC_1:def 17;
then b . x = FALSE by XBOOLEAN:def 3;
then 'not' (b . x) = TRUE by MARGREL1:11;
hence ('not' b) . x = TRUE by MARGREL1:def 19; :: thesis: verum

end;

thus ((All (('not' a),PA,G)) 'or' (All (('not' b),PA,G))) . z = ((All (('not' a),PA,G)) . z) 'or' ((All (('not' b),PA,G)) . z) by BVFUNC_1:def 4
.= ((All (('not' a),PA,G)) . z) 'or' TRUE by A1, A18, BVFUNC_1:def 16
.= TRUE by BINARITH:10 ; :: thesis: verum

end;

end;

end;

hence 'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) = (All (('not' a),PA,G)) 'or' (All (('not' b),PA,G)) by A4, BVFUNC_1:15; :: thesis: verum