let Y be non empty set ; :: thesis: for a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds 'not' (B_INF a,PA) = B_SUP ('not' a),PA
let a be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds 'not' (B_INF a,PA) = B_SUP ('not' a),PA
let PA be a_partition of Y; :: thesis: 'not' (B_INF a,PA) = B_SUP ('not' a),PA
consider k3 being Function such that
A1: 'not' (B_INF a,PA) = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: B_SUP ('not' a),PA = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for y being Element of Y holds ('not' (B_INF a,PA)) . y = (B_SUP ('not' a),PA) . y

proof

let y be Element of Y; :: thesis: ('not' (B_INF a,PA)) . y = (B_SUP ('not' a),PA) . y

A5: now

assume that
A6: for x being Element of Y st x in EqClass y,PA holds
a . x = TRUE and
A7: ex x being Element of Y st
( x in EqClass y,PA & ('not' a) . x = TRUE ) ; :: thesis: contradiction
consider x1 being Element of Y such that
A8: x1 in EqClass y,PA and
A9: ('not' a) . x1 = TRUE by A7;
('not' ('not' a)) . x1 = 'not' TRUE by A9, MARGREL1:def 20;
hence contradiction by A6, A8; :: thesis: verum

end;

A10: now

assume that
A11: ex x being Element of Y st
( x in EqClass y,PA & not a . x = TRUE ) and
A12: for x being Element of Y holds
( not x in EqClass y,PA or not ('not' a) . x = TRUE ) ; :: thesis: contradiction
consider x1 being Element of Y such that
A13: x1 in EqClass y,PA and
A14: a . x1 <> TRUE by A11;
a . x1 = FALSE by A14, XBOOLEAN:def 3;
then ('not' a) . x1 = 'not' FALSE by MARGREL1:def 20;
hence contradiction by A12, A13; :: thesis: verum

end;

A15: now

assume that
A16: ex x being Element of Y st
( x in EqClass y,PA & not a . x = TRUE ) and
A17: ex x being Element of Y st
( x in EqClass y,PA & ('not' a) . x = TRUE ) ; :: thesis: ('not' (B_INF a,PA)) . y = (B_SUP ('not' a),PA) . y
(B_INF a,PA) . y = FALSE by A16, Def19;
then ('not' (B_INF a,PA)) . y = 'not' FALSE by MARGREL1:def 20;
hence ('not' (B_INF a,PA)) . y = (B_SUP ('not' a),PA) . y by A17, Def20; :: thesis: verum

end;

now

assume that
A18: for x being Element of Y st x in EqClass y,PA holds
a . x = TRUE and
A19: for x being Element of Y holds
( not x in EqClass y,PA or not ('not' a) . x = TRUE ) ; :: thesis: ('not' (B_INF a,PA)) . y = (B_SUP ('not' a),PA) . y
(B_INF a,PA) . y = TRUE by A18, Def19;
then ('not' (B_INF a,PA)) . y = 'not' TRUE by MARGREL1:def 20;
hence ('not' (B_INF a,PA)) . y = (B_SUP ('not' a),PA) . y by A19, Def20; :: thesis: verum

end;

hence ('not' (B_INF a,PA)) . y = (B_SUP ('not' a),PA) . y by A5, A15, A10; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence 'not' (B_INF a,PA) = B_SUP ('not' a),PA by A1, A2, A3, A4, FUNCT_1:9; :: thesis: verum