let Y be non empty set ; :: thesis: for a, b, c, d being Function of Y,BOOLEAN holds ((a 'or' b) 'or' c) 'imp' d = ((a 'imp' d) '&' (b 'imp' d)) '&' (c 'imp' d)
let a, b, c, d be Function of Y,BOOLEAN; :: thesis: ((a 'or' b) 'or' c) 'imp' d = ((a 'imp' d) '&' (b 'imp' d)) '&' (c 'imp' d)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: K11((((a 'or' b) 'or' c) 'imp' d),x) = K11((((a 'imp' d) '&' (b 'imp' d)) '&' (c 'imp' d)),x)
(((a 'imp' d) '&' (b 'imp' d)) '&' (c 'imp' d)) . x = (((a 'imp' d) '&' (b 'imp' d)) . x) '&' ((c 'imp' d) . x) by MARGREL1:def 20
.= (((a 'imp' d) . x) '&' ((b 'imp' d) . x)) '&' ((c 'imp' d) . x) by MARGREL1:def 20
.= ((('not' (a . x)) 'or' (d . x)) '&' ((b 'imp' d) . x)) '&' ((c 'imp' d) . x) by BVFUNC_1:def 8
.= ((('not' (a . x)) 'or' (d . x)) '&' (('not' (b . x)) 'or' (d . x))) '&' ((c 'imp' d) . x) by BVFUNC_1:def 8
.= (((d . x) 'or' ('not' (a . x))) '&' (('not' (b . x)) 'or' (d . x))) '&' (('not' (c . x)) 'or' (d . x)) by BVFUNC_1:def 8
.= (('not' ((a . x) 'or' (b . x))) 'or' (d . x)) '&' (('not' (c . x)) 'or' (d . x)) by XBOOLEAN:9
.= ((d . x) 'or' ('not' ((a 'or' b) . x))) '&' (('not' (c . x)) 'or' (d . x)) by BVFUNC_1:def 4
.= ('not' (((a 'or' b) . x) 'or' (c . x))) 'or' (d . x) by XBOOLEAN:9
.= ('not' (((a 'or' b) 'or' c) . x)) 'or' (d . x) by BVFUNC_1:def 4
.= (((a 'or' b) 'or' c) 'imp' d) . x by BVFUNC_1:def 8 ;
hence K11((((a 'or' b) 'or' c) 'imp' d),x) = K11((((a 'imp' d) '&' (b 'imp' d)) '&' (c 'imp' d)),x) ; :: thesis: verum