let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a, b being Element of Funcs 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 Element of Funcs 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 Element of Funcs 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 15 :: 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 7;

per cases ( (All ('not' a),PA,G) . z = TRUE or (All ('not' b),PA,G) . z = TRUE ) by A6, A5, BINARITH:7;

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

A8: now

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 19;
hence contradiction by A7, BVFUNC_2:def 9; :: thesis: verum

end;

A9: now

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 20;
hence a . x <> TRUE by MARGREL1:41; :: 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 20
.= 'not' (((Ex a,PA,G) . z) '&' ((Ex b,PA,G) . z)) by MARGREL1:def 21
.= 'not' (FALSE '&' ((Ex b,PA,G) . z)) by A3, A9, BVFUNC_1:def 20
.= 'not' FALSE by MARGREL1:45
.= TRUE by MARGREL1:41 ; :: 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

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 19;
hence contradiction by A10, BVFUNC_2:def 9; :: thesis: verum

end;

A12: now

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 20;
hence b . x <> TRUE by MARGREL1:41; :: 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 20
.= 'not' (((Ex a,PA,G) . z) '&' ((Ex b,PA,G) . z)) by MARGREL1:def 21
.= 'not' (((Ex a,PA,G) . z) '&' FALSE ) by A2, A12, BVFUNC_1:def 20
.= 'not' FALSE by MARGREL1:45
.= TRUE by MARGREL1:41 ; :: 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 15 :: 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 20;
then ((Ex a,PA,G) '&' (Ex b,PA,G)) . z = FALSE by MARGREL1:41;
then A14: ((Ex a,PA,G) . z) '&' ((Ex b,PA,G) . z) = FALSE by MARGREL1:def 21;

per cases ( (Ex a,PA,G) . z = FALSE or (Ex b,PA,G) . z = FALSE ) by A14, MARGREL1:45;

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

A16: now

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 20;
then a . x = FALSE by XBOOLEAN:def 3;
then 'not' (a . x) = TRUE by MARGREL1:41;
hence ('not' a) . x = TRUE by MARGREL1:def 20; :: 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 7
.= TRUE 'or' ((All ('not' b),PA,G) . z) by A13, A16, BVFUNC_1:def 19
.= TRUE by BINARITH:19 ; :: 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

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 20;
then b . x = FALSE by XBOOLEAN:def 3;
then 'not' (b . x) = TRUE by MARGREL1:41;
hence ('not' b) . x = TRUE by MARGREL1:def 20; :: 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 7
.= ((All ('not' a),PA,G) . z) 'or' TRUE by A1, A18, BVFUNC_1:def 19
.= TRUE by BINARITH:19 ; :: 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_1:18; :: thesis: verum