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