let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a being Function of Y,BOOLEAN
for PA being a_partition of Y holds 'not' (Ex (a,PA,G)) = All (('not' a),PA,G)
let G be Subset of (PARTITIONS Y); :: thesis: for a being Function of Y,BOOLEAN
for PA being a_partition of Y holds 'not' (Ex (a,PA,G)) = All (('not' a),PA,G)
let a be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds 'not' (Ex (a,PA,G)) = All (('not' a),PA,G)
let PA be a_partition of Y; :: thesis: 'not' (Ex (a,PA,G)) = All (('not' a),PA,G)
let z be Element of Y; :: according to FUNCT_2:def 8 :: thesis: ('not' (Ex (a,PA,G))) . z = (All (('not' a),PA,G)) . z

per cases ( ( ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
('not' a) . x = TRUE ) & ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) ) or ( ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
('not' a) . x = TRUE ) & ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ) or ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not ('not' a) . 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 ('not' a) . x = TRUE ) & ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ) ) ;

suppose A1: ( ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
('not' a) . x = TRUE ) & ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) ) ; :: thesis: ('not' (Ex (a,PA,G))) . z = (All (('not' a),PA,G)) . z

then consider x1 being Element of Y such that
A2: x1 in EqClass (z,(CompF (PA,G))) and
A3: a . x1 = TRUE ;
('not' a) . x1 = 'not' TRUE by A3, MARGREL1:def 19;
hence ('not' (Ex (a,PA,G))) . z = (All (('not' a),PA,G)) . z by A1, A2, MARGREL1:11; :: thesis: verum

end;

suppose A4: ( ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
('not' a) . x = TRUE ) & ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ) ; :: thesis: ('not' (Ex (a,PA,G))) . z = (All (('not' a),PA,G)) . z

then 'not' ((B_SUP (a,(CompF (PA,G)))) . z) = TRUE by BVFUNC_1:def 17, MARGREL1:11;
then ('not' (B_SUP (a,(CompF (PA,G))))) . z = TRUE by MARGREL1:def 19;
hence ('not' (Ex (a,PA,G))) . z = (All (('not' a),PA,G)) . z by A4, BVFUNC_1:def 16; :: thesis: verum

end;

suppose A5: ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not ('not' a) . x = TRUE ) & ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) ) ; :: thesis: ('not' (Ex (a,PA,G))) . z = (All (('not' a),PA,G)) . z

then (B_SUP (a,(CompF (PA,G)))) . z = TRUE by BVFUNC_1:def 17;
then A6: ('not' (B_SUP (a,(CompF (PA,G))))) . z = 'not' TRUE by MARGREL1:def 19;
(B_INF (('not' a),(CompF (PA,G)))) . z = FALSE by A5, BVFUNC_1:def 16;
hence ('not' (Ex (a,PA,G))) . z = (All (('not' a),PA,G)) . z by A6, MARGREL1:11; :: thesis: verum

end;

suppose A7: ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not ('not' a) . x = TRUE ) & ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ) ; :: thesis: ('not' (Ex (a,PA,G))) . z = (All (('not' a),PA,G)) . z

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 = FALSE by A9, XBOOLEAN:def 3;
then 'not' (a . x1) = FALSE by MARGREL1:def 19;
hence ('not' (Ex (a,PA,G))) . z = (All (('not' a),PA,G)) . z by A7, A8, MARGREL1:11; :: thesis: verum

end;

end;