let Y be non empty set ; :: thesis: for a, b being Element of Funcs Y,BOOLEAN holds a '&' (a 'imp' b) '<' b
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: a '&' (a 'imp' b) '<' b
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (a '&' (a 'imp' b)) . z = TRUE or b . z = TRUE )
assume A1: (a '&' (a 'imp' b)) . z = TRUE ; :: thesis: b . z = TRUE
A2: (a '&' (a 'imp' b)) . z = (a . z) '&' ((a 'imp' b) . z) by MARGREL1:def 21
.= (a . z) '&' ((('not' a) 'or' b) . z) by BVFUNC_4:8
.= (a . z) '&' ((('not' a) . z) 'or' (b . z)) by BVFUNC_1:def 7
.= ((a . z) '&' (('not' a) . z)) 'or' ((a . z) '&' (b . z)) by XBOOLEAN:8
.= ((a . z) '&' ('not' (a . z))) 'or' ((a . z) '&' (b . z)) by MARGREL1:def 20
.= FALSE 'or' ((a . z) '&' (b . z)) by XBOOLEAN:138
.= (a . z) '&' (b . z) ;

now

assume b . z <> TRUE ; :: thesis: contradiction
then (a . z) '&' (b . z) = FALSE '&' (a . z) by XBOOLEAN:def 3
.= FALSE ;
hence contradiction by A1, A2; :: thesis: verum

end;

hence b . z = TRUE ; :: thesis: verum