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' ((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 Element of Funcs 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 Element of Funcs 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 15 :: 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 7;
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:7;

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

now

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 20;
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 20;
then A10: a . x1 = FALSE by MARGREL1:41;
thus ('not' ((All a,PA,G) '&' (All b,PA,G))) . z = 'not' (((All a,PA,G) '&' (All b,PA,G)) . z) by MARGREL1:def 20
.= 'not' (((All a,PA,G) . z) '&' ((All b,PA,G) . z)) by MARGREL1:def 21
.= 'not' (FALSE '&' ((All b,PA,G) . z)) by A2, A8, A10, BVFUNC_1:def 19
.= 'not' FALSE by MARGREL1:45
.= TRUE by MARGREL1:41 ; :: 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

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 20;
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 20;
then A14: b . x1 = FALSE by MARGREL1:41;
thus ('not' ((All a,PA,G) '&' (All b,PA,G))) . z = 'not' (((All a,PA,G) '&' (All b,PA,G)) . z) by MARGREL1:def 20
.= 'not' (((All a,PA,G) . z) '&' ((All b,PA,G) . z)) by MARGREL1:def 21
.= 'not' (((All a,PA,G) . z) '&' FALSE ) by A1, A12, A14, BVFUNC_1:def 19
.= 'not' FALSE by MARGREL1:45
.= TRUE by MARGREL1:41 ; :: 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 15 :: 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 20;
((All a,PA,G) '&' (All b,PA,G)) . z = ((All a,PA,G) . z) '&' ((All b,PA,G) . z) by MARGREL1:def 21;
then A16: ((All a,PA,G) . z) '&' ((All b,PA,G) . z) = FALSE by A15, MARGREL1:41;

per cases ( (All a,PA,G) . z = FALSE or (All b,PA,G) . z = FALSE ) by A16, MARGREL1:45;

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 19;
a . x1 = FALSE by A18, XBOOLEAN:def 3;
then 'not' (a . x1) = TRUE by MARGREL1:41;
then ('not' a) . x1 = TRUE by MARGREL1:def 20;
then (B_SUP ('not' a),(CompF PA,G)) . z = TRUE by A17, BVFUNC_1:def 20;
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 7
.= TRUE by BINARITH:19 ;
:: 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 19;
b . x1 = FALSE by A20, XBOOLEAN:def 3;
then 'not' (b . x1) = TRUE by MARGREL1:41;
then ('not' b) . x1 = TRUE by MARGREL1:def 20;
then (B_SUP ('not' b),(CompF PA,G)) . z = TRUE by A19, BVFUNC_1:def 20;
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 7
.= TRUE by BINARITH:19 ;
:: 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_1:18; :: thesis: verum