let Y be non empty set ; :: thesis: for a being Element of Funcs (Y,BOOLEAN) holds
( O_el Y '<' a & a '<' I_el Y )
let a be Element of Funcs (Y,BOOLEAN); :: thesis: ( O_el Y '<' a & a '<' I_el Y )
consider k3 being Function such that
A1: (O_el Y) 'imp' a = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: I_el Y = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for x being Element of Y holds ((O_el Y) 'imp' a) . x = (I_el Y) . x

proof

let x be Element of Y; :: thesis: ((O_el Y) 'imp' a) . x = (I_el Y) . x
((O_el Y) 'imp' a) . x = ('not' ((O_el Y) . x)) 'or' (a . x) by Def11;
then ((O_el Y) 'imp' a) . x = TRUE 'or' (a . x) by Def13;
hence ((O_el Y) 'imp' a) . x = (I_el Y) . x by Def14; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
then A5: k3 = k4 by A2, A4, FUNCT_1:2;
consider k4 being Function such that
A6: I_el Y = k4 and
A7: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
consider k3 being Function such that
A8: a 'imp' (I_el Y) = k3 and
A9: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
for x being Element of Y holds (a 'imp' (I_el Y)) . x = (I_el Y) . x

proof

let x be Element of Y; :: thesis: (a 'imp' (I_el Y)) . x = (I_el Y) . x
(a 'imp' (I_el Y)) . x = ('not' (a . x)) 'or' ((I_el Y) . x) by Def11;
then (a 'imp' (I_el Y)) . x = ('not' (a . x)) 'or' TRUE by Def14;
hence (a 'imp' (I_el Y)) . x = (I_el Y) . x by Def14; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A8, A6;
then a 'imp' (I_el Y) = I_el Y by A8, A9, A6, A7, FUNCT_1:2;
hence ( O_el Y '<' a & a '<' I_el Y ) by A1, A3, A5, Th19; :: thesis: verum