let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds a 'imp' b '<' a 'imp' (b 'or' c)
let a, b, c be Function of Y,BOOLEAN; :: thesis: a 'imp' b '<' a 'imp' (b 'or' c)
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (a 'imp' b) . z = TRUE or (a 'imp' (b 'or' c)) . z = TRUE )
A1: (a 'imp' b) . z = (('not' a) 'or' b) . z by BVFUNC_4:8
.= (('not' a) . z) 'or' (b . z) by BVFUNC_1:def 4 ;
assume A2: (a 'imp' b) . z = TRUE ; :: thesis: (a 'imp' (b 'or' c)) . z = TRUE

now :: thesis: not (a 'imp' (b 'or' c)) . z <> TRUE

assume A3: (a 'imp' (b 'or' c)) . z <> TRUE ; :: thesis: contradiction
(a 'imp' (b 'or' c)) . z = (('not' a) 'or' (b 'or' c)) . z by BVFUNC_4:8
.= (('not' a) . z) 'or' ((b 'or' c) . z) by BVFUNC_1:def 4
.= (('not' a) . z) 'or' ((b . z) 'or' (c . z)) by BVFUNC_1:def 4
.= ((('not' a) . z) 'or' (b . z)) 'or' (c . z)
.= TRUE by A2, A1 ;
hence contradiction by A3; :: thesis: verum

end;

hence (a 'imp' (b 'or' c)) . z = TRUE ; :: thesis: verum