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

proof

let x be Element of Y; :: thesis: (a '&' (a 'imp' b)) . x = (a '&' b) . x
(a '&' (a 'imp' b)) . x = (a . x) '&' ((a 'imp' b) . x) by MARGREL1:def 21
.= (a . x) '&' (('not' (a . x)) 'or' (b . x)) by BVFUNC_1:def 11
.= ((a . x) '&' ('not' (a . x))) 'or' ((a . x) '&' (b . x)) by XBOOLEAN:8
.= FALSE 'or' ((a . x) '&' (b . x)) by XBOOLEAN:138
.= (a . x) '&' (b . x) by BINARITH:7
.= (a '&' b) . x by MARGREL1:def 21 ;
hence (a '&' (a 'imp' b)) . x = (a '&' b) . x ; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence a '&' (a 'imp' b) = a '&' b by A1, A2, A3, A4, FUNCT_1:9; :: thesis: verum