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

proof

let x be Element of Y; :: thesis: (a 'nand' b) . x = ('not' (a '&' b)) . x
thus ('not' (a '&' b)) . x = 'not' ((a '&' b) . x) by MARGREL1:def 19
.= 'not' ((a . x) '&' (b . x)) by MARGREL1:def 20
.= 'not' ('not' ((a . x) 'nand' (b . x))) by BVFUNC_1:46
.= (a 'nand' b) . x by Def3 ; :: thesis: verum

end;

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