let Y be non empty set ; :: thesis: for a, b being Element of Funcs Y,BOOLEAN holds a '&' (a 'or' b) = a
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: a '&' (a 'or' b) = a
consider k3 being Function such that
A1: a '&' (a 'or' b) = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: a = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for x being Element of Y holds (a '&' (a 'or' b)) . x = a . x

proof

let x be Element of Y; :: thesis: (a '&' (a 'or' b)) . x = a . x
A5: (a '&' (a 'or' b)) . x = (a . x) '&' ((a 'or' b) . x) by MARGREL1:def 21
.= (a . x) '&' ((a . x) 'or' (b . x)) by BVFUNC_1:def 7 ;

now

per cases ( a . x = TRUE or a . x = FALSE ) by XBOOLEAN:def 3;

case a . x = TRUE ; :: thesis: (a '&' (a 'or' b)) . x = a . x

hence (a '&' (a 'or' b)) . x = a . x by A5; :: thesis: verum

end;

case a . x = FALSE ; :: thesis: (a '&' (a 'or' b)) . x = a . x

hence (a '&' (a 'or' b)) . x = a . x by A5; :: thesis: verum

end;

end;

end;

hence (a '&' (a 'or' b)) . x = a . x ; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence a '&' (a 'or' b) = a by A1, A2, A3, A4, FUNCT_1:9; :: thesis: verum