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

proof

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ((Ex (('not' a),PA,G)) 'or' (Ex (('not' b),PA,G))) . z = TRUE or ('not' ((All (a,PA,G)) '&' (All (b,PA,G)))) . z = TRUE )
A4: ((Ex (('not' a),PA,G)) 'or' (Ex (('not' b),PA,G))) . z = ((Ex (('not' a),PA,G)) . z) 'or' ((Ex (('not' b),PA,G)) . z) by BVFUNC_1:def 4;
A5: ( (Ex (('not' b),PA,G)) . z = TRUE or (Ex (('not' b),PA,G)) . z = FALSE ) by XBOOLEAN:def 3;
assume A6: ((Ex (('not' a),PA,G)) 'or' (Ex (('not' b),PA,G))) . z = TRUE ; :: thesis: ('not' ((All (a,PA,G)) '&' (All (b,PA,G)))) . z = TRUE

per cases ( (Ex (('not' a),PA,G)) . z = TRUE or (Ex (('not' b),PA,G)) . z = TRUE ) by A6, A4, A5, BINARITH:3;

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

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

assume for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not ('not' a) . x = TRUE ) ; :: thesis: contradiction
then (B_SUP (('not' a),(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 17;
hence contradiction by A7, BVFUNC_2:def 10; :: thesis: verum

end;

then consider x1 being Element of Y such that
A8: x1 in EqClass (z,(CompF (PA,G))) and
A9: ('not' a) . x1 = TRUE ;
'not' (a . x1) = TRUE by A9, MARGREL1:def 19;
then A10: a . x1 = FALSE by MARGREL1:11;
thus ('not' ((All (a,PA,G)) '&' (All (b,PA,G)))) . z = 'not' (((All (a,PA,G)) '&' (All (b,PA,G))) . z) by MARGREL1:def 19
.= 'not' (((All (a,PA,G)) . z) '&' ((All (b,PA,G)) . z)) by MARGREL1:def 20
.= 'not' (FALSE '&' ((All (b,PA,G)) . z)) by A2, A8, A10, BVFUNC_1:def 16
.= 'not' FALSE by MARGREL1:12
.= TRUE by MARGREL1:11 ; :: thesis: verum

end;

suppose A11: (Ex (('not' b),PA,G)) . z = TRUE ; :: thesis: ('not' ((All (a,PA,G)) '&' (All (b,PA,G)))) . z = TRUE

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

assume for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not ('not' b) . x = TRUE ) ; :: thesis: contradiction
then (B_SUP (('not' b),(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 17;
hence contradiction by A11, BVFUNC_2:def 10; :: thesis: verum

end;

then consider x1 being Element of Y such that
A12: x1 in EqClass (z,(CompF (PA,G))) and
A13: ('not' b) . x1 = TRUE ;
'not' (b . x1) = TRUE by A13, MARGREL1:def 19;
then A14: b . x1 = FALSE by MARGREL1:11;
thus ('not' ((All (a,PA,G)) '&' (All (b,PA,G)))) . z = 'not' (((All (a,PA,G)) '&' (All (b,PA,G))) . z) by MARGREL1:def 19
.= 'not' (((All (a,PA,G)) . z) '&' ((All (b,PA,G)) . z)) by MARGREL1:def 20
.= 'not' (((All (a,PA,G)) . z) '&' FALSE) by A1, A12, A14, BVFUNC_1:def 16
.= 'not' FALSE by MARGREL1:12
.= TRUE by MARGREL1:11 ; :: thesis: verum

end;

end;

end;

'not' ((All (a,PA,G)) '&' (All (b,PA,G))) '<' (Ex (('not' a),PA,G)) 'or' (Ex (('not' b),PA,G))

proof

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ('not' ((All (a,PA,G)) '&' (All (b,PA,G)))) . z = TRUE or ((Ex (('not' a),PA,G)) 'or' (Ex (('not' b),PA,G))) . z = TRUE )
assume ('not' ((All (a,PA,G)) '&' (All (b,PA,G)))) . z = TRUE ; :: thesis: ((Ex (('not' a),PA,G)) 'or' (Ex (('not' b),PA,G))) . z = TRUE
then A15: 'not' (((All (a,PA,G)) '&' (All (b,PA,G))) . z) = TRUE by MARGREL1:def 19;
((All (a,PA,G)) '&' (All (b,PA,G))) . z = ((All (a,PA,G)) . z) '&' ((All (b,PA,G)) . z) by MARGREL1:def 20;
then A16: ((All (a,PA,G)) . z) '&' ((All (b,PA,G)) . z) = FALSE by A15, MARGREL1:11;

per cases ( (All (a,PA,G)) . z = FALSE or (All (b,PA,G)) . z = FALSE ) by A16, MARGREL1:12;

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

then consider x1 being Element of Y such that
A17: x1 in EqClass (z,(CompF (PA,G))) and
A18: a . x1 <> TRUE by A2, BVFUNC_1:def 16;
a . x1 = FALSE by A18, XBOOLEAN:def 3;
then 'not' (a . x1) = TRUE by MARGREL1:11;
then ('not' a) . x1 = TRUE by MARGREL1:def 19;
then (B_SUP (('not' a),(CompF (PA,G)))) . z = TRUE by A17, BVFUNC_1:def 17;
then (Ex (('not' a),PA,G)) . z = TRUE by BVFUNC_2:def 10;
hence ((Ex (('not' a),PA,G)) 'or' (Ex (('not' b),PA,G))) . z = TRUE 'or' ((Ex (('not' b),PA,G)) . z) by BVFUNC_1:def 4
.= TRUE by BINARITH:10 ;
:: thesis: verum

end;

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

then consider x1 being Element of Y such that
A19: x1 in EqClass (z,(CompF (PA,G))) and
A20: b . x1 <> TRUE by A1, BVFUNC_1:def 16;
b . x1 = FALSE by A20, XBOOLEAN:def 3;
then 'not' (b . x1) = TRUE by MARGREL1:11;
then ('not' b) . x1 = TRUE by MARGREL1:def 19;
then (B_SUP (('not' b),(CompF (PA,G)))) . z = TRUE by A19, BVFUNC_1:def 17;
then (Ex (('not' b),PA,G)) . z = TRUE by BVFUNC_2:def 10;
hence ((Ex (('not' a),PA,G)) 'or' (Ex (('not' b),PA,G))) . z = ((Ex (('not' a),PA,G)) . z) 'or' TRUE by BVFUNC_1:def 4
.= TRUE by BINARITH:10 ;
:: thesis: verum

end;

end;

end;

hence 'not' ((All (a,PA,G)) '&' (All (b,PA,G))) = (Ex (('not' a),PA,G)) 'or' (Ex (('not' b),PA,G)) by A3, BVFUNC_1:15; :: thesis: verum