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 )
A1: 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;

consider k3 being Function such that
A2: ( (O_el Y) 'imp' a = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A3: ( I_el Y = k4 & dom k4 = Y & rng k4 c= BOOLEAN ) by FUNCT_2:def 2;
for u being set st u in Y holds
k3 . u = k4 . u by A1, A2, A3;
then A4: k3 = k4 by A2, A3, FUNCT_1:9;
A5: 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;

consider k3 being Function such that
A6: ( a 'imp' (I_el Y) = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A7: ( I_el Y = k4 & dom k4 = Y & rng k4 c= BOOLEAN ) by FUNCT_2:def 2;
for u being set st u in Y holds
k3 . u = k4 . u by A5, A6, A7;
then a 'imp' (I_el Y) = I_el Y by A6, A7, FUNCT_1:9;
hence ( O_el Y '<' a & a '<' I_el Y ) by A2, A3, A4, Th19; :: thesis: verum