let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds (a 'imp' b) '&' (('not' a) 'imp' b) = b
let a, b be Function of Y,BOOLEAN; :: thesis: (a 'imp' b) '&' (('not' a) 'imp' b) = b
let x be Element of Y; :: according to FUNCT_2:def 8 :: thesis: K11(((a 'imp' b) '&' (('not' a) 'imp' b)),x) = K11(b,x)
A1: ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = ((a 'imp' b) . x) '&' ((('not' a) 'imp' b) . x) by MARGREL1:def 20
.= (('not' (a . x)) 'or' (b . x)) '&' ((('not' a) 'imp' b) . x) by BVFUNC_1:def 8
.= (('not' (a . x)) 'or' (b . x)) '&' (('not' (('not' a) . x)) 'or' (b . x)) by BVFUNC_1:def 8
.= (('not' (a . x)) 'or' (b . x)) '&' ((a . x) 'or' (b . x)) by MARGREL1:def 19 ;

now :: thesis: ( ( a . x = TRUE & K11(((a 'imp' b) '&' (('not' a) 'imp' b)),x) = K11(b,x) ) or ( a . x = FALSE & K11(((a 'imp' b) '&' (('not' a) 'imp' b)),x) = K11(b,x) ) )

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

case a . x = TRUE ; :: thesis: K11(((a 'imp' b) '&' (('not' a) 'imp' b)),x) = K11(b,x)

then ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = (FALSE 'or' (b . x)) '&' (TRUE 'or' (b . x)) by A1
.= (FALSE 'or' (b . x)) '&' TRUE
.= TRUE '&' (b . x)
.= b . x ;
hence K11(((a 'imp' b) '&' (('not' a) 'imp' b)),x) = K11(b,x) ; :: thesis: verum

end;

case a . x = FALSE ; :: thesis: K11(((a 'imp' b) '&' (('not' a) 'imp' b)),x) = K11(b,x)

then ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = (TRUE 'or' (b . x)) '&' (FALSE 'or' (b . x)) by A1
.= TRUE '&' (FALSE 'or' (b . x))
.= TRUE '&' (b . x)
.= b . x ;
hence K11(((a 'imp' b) '&' (('not' a) 'imp' b)),x) = K11(b,x) ; :: thesis: verum

end;

end;

end;

hence K11(((a 'imp' b) '&' (('not' a) 'imp' b)),x) = K11(b,x) ; :: thesis: verum