let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds a 'imp' (b '&' c) = (a 'imp' b) '&' (a 'imp' c)
let a, b, c be Function of Y,BOOLEAN; :: thesis: a 'imp' (b '&' c) = (a 'imp' b) '&' (a 'imp' c)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: K11((a 'imp' (b '&' c)),x) = K11(((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 20
.= (('not' (a . x)) 'or' (b . x)) '&' ((a 'imp' c) . x) by BVFUNC_1:def 8
.= (('not' (a . x)) 'or' (b . x)) '&' (('not' (a . x)) 'or' (c . x)) by BVFUNC_1:def 8
.= ('not' (a . x)) 'or' ((b . x) '&' (c . x)) by XBOOLEAN:9
.= ('not' (a . x)) 'or' ((b '&' c) . x) by MARGREL1:def 20
.= (a 'imp' (b '&' c)) . x by BVFUNC_1:def 8 ;
hence K11((a 'imp' (b '&' c)),x) = K11(((a 'imp' b) '&' (a 'imp' c)),x) ; :: thesis: verum