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

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

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

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

then ((a 'imp' b) '&' (a 'imp' ('not' b))) . x = (('not' (a . x)) 'or' TRUE) '&' (('not' (a . x)) 'or' FALSE) by A1
.= (('not' (a . x)) 'or' TRUE) '&' ('not' (a . x))
.= TRUE '&' ('not' (a . x))
.= 'not' (a . x)
.= ('not' a) . x by MARGREL1:def 19 ;
hence K11(((a 'imp' b) '&' (a 'imp' ('not' b))),x) = K11(('not' a),x) ; :: thesis: verum

end;

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

then ((a 'imp' b) '&' (a 'imp' ('not' b))) . x = (('not' (a . x)) 'or' FALSE) '&' (('not' (a . x)) 'or' TRUE) by A1
.= ('not' (a . x)) '&' (('not' (a . x)) 'or' TRUE)
.= TRUE '&' ('not' (a . x))
.= 'not' (a . x)
.= ('not' a) . x by MARGREL1:def 19 ;
hence K11(((a 'imp' b) '&' (a 'imp' ('not' b))),x) = K11(('not' a),x) ; :: thesis: verum

end;

end;

end;

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