let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds a = ((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))
let a, b, c be Function of Y,BOOLEAN; :: thesis: a = ((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: K11(a,x) = K11((((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))),x)
(((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))) . x = ((((a '&' b) '&' (c 'or' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))) . x by BVFUNC_1:12
.= ((((a '&' b) '&' (I_el Y)) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))) . x by BVFUNC_4:6
.= (((a '&' b) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))) . x by BVFUNC_1:6
.= ((a '&' b) 'or' (((a '&' ('not' b)) '&' c) 'or' ((a '&' ('not' b)) '&' ('not' c)))) . x by BVFUNC_1:8
.= ((a '&' b) 'or' ((a '&' ('not' b)) '&' (c 'or' ('not' c)))) . x by BVFUNC_1:12
.= ((a '&' b) 'or' ((a '&' ('not' b)) '&' (I_el Y))) . x by BVFUNC_4:6
.= ((a '&' b) 'or' (a '&' ('not' b))) . x by BVFUNC_1:6
.= (a '&' (b 'or' ('not' b))) . x by BVFUNC_1:12
.= (a '&' (I_el Y)) . x by BVFUNC_4:6
.= a . x by BVFUNC_1:6 ;
hence K11(a,x) = K11((((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))),x) ; :: thesis: verum