let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds (a '&' b) 'imp' c = (a 'imp' c) 'or' (b 'imp' c)
let a, b, c be Function of Y,BOOLEAN; :: thesis: (a '&' b) 'imp' c = (a 'imp' c) 'or' (b 'imp' c)
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: K11(((a '&' b) 'imp' c),x) = K11(((a 'imp' c) 'or' (b 'imp' c)),x)
((a 'imp' c) 'or' (b 'imp' c)) . x = ((a 'imp' c) . x) 'or' ((b 'imp' c) . x) by BVFUNC_1:def 4
.= (('not' (a . x)) 'or' (c . x)) 'or' ((b 'imp' c) . x) by BVFUNC_1:def 8
.= (('not' (a . x)) 'or' (c . x)) 'or' (('not' (b . x)) 'or' (c . x)) by BVFUNC_1:def 8
.= (('not' (a . x)) 'or' ((c . x) 'or' ('not' (b . x)))) 'or' (c . x)
.= ((('not' (a . x)) 'or' ('not' (b . x))) 'or' (c . x)) 'or' (c . x)
.= (('not' (a . x)) 'or' ('not' (b . x))) 'or' ((c . x) 'or' (c . x)) by BINARITH:11
.= ('not' ((a '&' b) . x)) 'or' (c . x) by MARGREL1:def 20
.= ((a '&' b) 'imp' c) . x by BVFUNC_1:def 8 ;
hence K11(((a '&' b) 'imp' c),x) = K11(((a 'imp' c) 'or' (b 'imp' c)),x) ; :: thesis: verum