let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds ((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c)) = I_el Y
let a, b, c be Function of Y,BOOLEAN; :: thesis: ((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c)) = I_el Y
for x being Element of Y holds (((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c))) . x = TRUE

proof

let x be Element of Y; :: thesis: (((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c))) . x = TRUE
(((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c))) . x = ('not' (((a 'or' b) '&' c) . x)) 'or' (((a '&' c) 'or' (b '&' c)) . x) by BVFUNC_1:def 8
.= ('not' (((a 'or' b) . x) '&' (c . x))) 'or' (((a '&' c) 'or' (b '&' c)) . x) by MARGREL1:def 20
.= ('not' (((a . x) 'or' (b . x)) '&' (c . x))) 'or' (((a '&' c) 'or' (b '&' c)) . x) by BVFUNC_1:def 4
.= ('not' (((a . x) 'or' (b . x)) '&' (c . x))) 'or' (((a '&' c) . x) 'or' ((b '&' c) . x)) by BVFUNC_1:def 4
.= ('not' (((a . x) 'or' (b . x)) '&' (c . x))) 'or' (((a . x) '&' (c . x)) 'or' ((b '&' c) . x)) by MARGREL1:def 20
.= ('not' ((c . x) '&' ((a . x) 'or' (b . x)))) 'or' (((a . x) '&' (c . x)) 'or' ((b . x) '&' (c . x))) by MARGREL1:def 20
.= (((a . x) '&' (c . x)) 'or' ((b . x) '&' (c . x))) 'or' (('not' ((a . x) '&' (c . x))) '&' ('not' ((b . x) '&' (c . x)))) by XBOOLEAN:8
.= ((((b . x) '&' (c . x)) 'or' ((a . x) '&' (c . x))) 'or' ('not' ((a . x) '&' (c . x)))) '&' ((((a . x) '&' (c . x)) 'or' ((b . x) '&' (c . x))) 'or' ('not' ((b . x) '&' (c . x)))) by XBOOLEAN:9
.= (((b . x) '&' (c . x)) 'or' (((a . x) '&' (c . x)) 'or' ('not' ((a . x) '&' (c . x))))) '&' (((a . x) '&' (c . x)) 'or' (((b . x) '&' (c . x)) 'or' ('not' ((b . x) '&' (c . x)))))
.= (((b . x) '&' (c . x)) 'or' TRUE) '&' (((a . x) '&' (c . x)) 'or' (((b . x) '&' (c . x)) 'or' ('not' ((b . x) '&' (c . x))))) by XBOOLEAN:102
.= TRUE '&' (((a . x) '&' (c . x)) 'or' TRUE) by XBOOLEAN:102
.= TRUE ;
hence (((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c))) . x = TRUE ; :: thesis: verum

end;

hence ((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c)) = I_el Y by BVFUNC_1:def 11; :: thesis: verum