let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN st a '<' b 'imp' c holds
a '&' b '<' c
let a, b, c be Function of Y,BOOLEAN; :: thesis: ( a '<' b 'imp' c implies a '&' b '<' c )
assume A1: a '<' b 'imp' c ; :: thesis: a '&' b '<' c
for x being Element of Y holds ((a '&' b) 'imp' c) . x = TRUE

proof

let x be Element of Y; :: thesis: ((a '&' b) 'imp' c) . x = TRUE
A2: (a 'imp' (b 'imp' c)) . x = ('not' (a . x)) 'or' ((b 'imp' c) . x) by BVFUNC_1:def 8
.= ('not' (a . x)) 'or' (('not' (b . x)) 'or' (c . x)) by BVFUNC_1:def 8
.= (('not' (a . x)) 'or' ('not' (b . x))) 'or' (c . x) ;
A3: ((a '&' b) 'imp' c) . x = ('not' ((a '&' b) . x)) 'or' (c . x) by BVFUNC_1:def 8
.= (('not' (a . x)) 'or' ('not' (b . x))) 'or' (c . x) by MARGREL1:def 20 ;
a 'imp' (b 'imp' c) = I_el Y by A1, BVFUNC_1:16;
hence ((a '&' b) 'imp' c) . x = TRUE by A2, A3, BVFUNC_1:def 11; :: thesis: verum

end;

then (a '&' b) 'imp' c = I_el Y by BVFUNC_1:def 11;
hence a '&' b '<' c by BVFUNC_1:16; :: thesis: verum