let Y be non empty set ; :: thesis: for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'or' b) 'imp' c
let a, b, c be Function of Y,BOOLEAN; :: thesis: (a 'imp' b) '&' (b 'imp' c) '<' (a 'or' b) 'imp' c
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ((a 'imp' b) '&' (b 'imp' c)) . z = TRUE or ((a 'or' b) 'imp' c) . z = TRUE )
A1: ((a 'imp' b) '&' (b 'imp' c)) . z = ((a 'imp' b) . z) '&' ((b 'imp' c) . z) by MARGREL1:def 20
.= ((('not' a) 'or' b) . z) '&' ((b 'imp' c) . z) by BVFUNC_4:8
.= ((('not' a) 'or' b) . z) '&' ((('not' b) 'or' c) . z) by BVFUNC_4:8
.= ((('not' a) . z) 'or' (b . z)) '&' ((('not' b) 'or' c) . z) by BVFUNC_1:def 4
.= ((('not' a) . z) 'or' (b . z)) '&' ((('not' b) . z) 'or' (c . z)) by BVFUNC_1:def 4 ;
assume A2: ((a 'imp' b) '&' (b 'imp' c)) . z = TRUE ; :: thesis: ((a 'or' b) 'imp' c) . z = TRUE

now :: thesis: ( ((a 'or' b) 'imp' c) . z <> TRUE implies ((a 'or' b) 'imp' c) . z = TRUE )

A3: ( c . z = TRUE or c . z = FALSE ) by XBOOLEAN:def 3;
assume A4: ((a 'or' b) 'imp' c) . z <> TRUE ; :: thesis: ((a 'or' b) 'imp' c) . z = TRUE
A5: ((a 'or' b) 'imp' c) . z = (('not' (a 'or' b)) 'or' c) . z by BVFUNC_4:8
.= ((('not' a) '&' ('not' b)) 'or' c) . z by BVFUNC_1:13
.= ((('not' a) '&' ('not' b)) . z) 'or' (c . z) by BVFUNC_1:def 4
.= ((('not' a) . z) '&' (('not' b) . z)) 'or' (c . z) by MARGREL1:def 20 ;
A6: ( (('not' a) . z) '&' (('not' b) . z) = TRUE or (('not' a) . z) '&' (('not' b) . z) = FALSE ) by XBOOLEAN:def 3;

now :: thesis: ( ( ('not' a) . z = FALSE & ((a 'or' b) 'imp' c) . z = TRUE ) or ( ('not' b) . z = FALSE & ((a 'or' b) 'imp' c) . z = TRUE ) )

per cases ( ('not' a) . z = FALSE or ('not' b) . z = FALSE ) by A4, A5, A6, MARGREL1:12;

case ('not' a) . z = FALSE ; :: thesis: ((a 'or' b) 'imp' c) . z = TRUE

then ((('not' a) . z) 'or' (b . z)) '&' ((('not' b) . z) 'or' (c . z)) = (b . z) '&' ('not' (b . z)) by A4, A5, A3, MARGREL1:def 19
.= FALSE by XBOOLEAN:138 ;
hence ((a 'or' b) 'imp' c) . z = TRUE by A2, A1; :: thesis: verum

end;

case ('not' b) . z = FALSE ; :: thesis: ((a 'or' b) 'imp' c) . z = TRUE

then ((('not' a) . z) 'or' (b . z)) '&' ((('not' b) . z) 'or' (c . z)) = ((('not' a) . z) 'or' (b . z)) '&' FALSE by A4, A5, XBOOLEAN:def 3
.= FALSE ;
hence ((a 'or' b) 'imp' c) . z = TRUE by A2, A1; :: thesis: verum

end;

end;

end;

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

end;

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