let Y be non empty set ; :: thesis: for a being Function of Y,BOOLEAN holds
( O_el Y '<' a & a '<' I_el Y )
let a be Function of Y,BOOLEAN; :: thesis: ( O_el Y '<' a & a '<' I_el Y )
A1: (O_el Y) 'imp' a = I_el Y

proof

let x be Element of Y; :: according to FUNCT_2:def 8 :: 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 Def8;
then ((O_el Y) 'imp' a) . x = TRUE 'or' (a . x) by Def10;
hence ((O_el Y) 'imp' a) . x = (I_el Y) . x by Def11; :: thesis: verum

end;

a 'imp' (I_el Y) = I_el Y

proof

let x be Element of Y; :: according to FUNCT_2:def 8 :: 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 Def8;
then (a 'imp' (I_el Y)) . x = ('not' (a . x)) 'or' TRUE by Def11;
hence (a 'imp' (I_el Y)) . x = (I_el Y) . x by Def11; :: thesis: verum

end;

hence ( O_el Y '<' a & a '<' I_el Y ) by A1, Th15; :: thesis: verum