let Y be non empty set ; :: thesis: for a, b, c being Element of Funcs (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 Element of Funcs (Y,BOOLEAN); :: thesis: a = ((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))
consider k3 being Function such that
A1: a = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: ((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c)) = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for x being Element of Y holds a . x = (((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))) . x

proof

let x be Element of Y; :: thesis: a . 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' ((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 a . x = (((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))) . x ; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence a = ((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c)) by A1, A2, A3, A4, FUNCT_1:2; :: thesis: verum