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

proof

let x be Element of Y; :: thesis: (a 'imp' (b '&' c)) . x = ((a 'imp' b) '&' (a 'imp' c)) . x
((a 'imp' b) '&' (a 'imp' c)) . x = ((a 'imp' b) . x) '&' ((a 'imp' c) . x) by MARGREL1:def 21
.= (('not' (a . x)) 'or' (b . x)) '&' ((a 'imp' c) . x) by BVFUNC_1:def 11
.= (('not' (a . x)) 'or' (b . x)) '&' (('not' (a . x)) 'or' (c . x)) by BVFUNC_1:def 11
.= ('not' (a . x)) 'or' ((b . x) '&' (c . x)) by XBOOLEAN:9
.= ('not' (a . x)) 'or' ((b '&' c) . x) by MARGREL1:def 21
.= (a 'imp' (b '&' c)) . x by BVFUNC_1:def 11 ;
hence (a 'imp' (b '&' c)) . x = ((a 'imp' b) '&' (a 'imp' c)) . x ; :: thesis: verum

end;

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