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: ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = b . x
A5: ((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 & ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = b . x ) or ( a . x = FALSE & ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = b . x ) )

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

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

then ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = (FALSE 'or' (b . x)) '&' (TRUE 'or' (b . x)) by A5, MARGREL1:11
.= (FALSE 'or' (b . x)) '&' TRUE by BINARITH:10
.= TRUE '&' (b . x) by BINARITH:3
.= b . x by MARGREL1:14 ;
hence ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = b . x ; :: thesis: verum

end;

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

then ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = (TRUE 'or' (b . x)) '&' (FALSE 'or' (b . x)) by A5, MARGREL1:11
.= TRUE '&' (FALSE 'or' (b . x)) by BINARITH:10
.= TRUE '&' (b . x) by BINARITH:3
.= b . x by MARGREL1:14 ;
hence ((a 'imp' b) '&' (('not' a) 'imp' b)) . x = b . x ; :: thesis: verum

end;

end;

end;

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