let A, B be non empty preBoolean set ; :: thesis: for a, b, c being Element of [:A,B:] holds a /\ (b \/ c) = (a /\ b) \/ (a /\ c)
let a, b, c be Element of [:A,B:]; :: thesis: a /\ (b \/ c) = (a /\ b) \/ (a /\ c)

now

thus (a /\ (b \/ c)) `1 = (a `1 ) /\ ((b \/ c) `1 ) by MCART_1:7
.= (a `1 ) /\ ((b `1 ) \/ (c `1 )) by MCART_1:7
.= ((a `1 ) /\ (b `1 )) \/ ((a `1 ) /\ (c `1 )) by XBOOLE_1:23
.= ((a `1 ) /\ (b `1 )) \/ ((a /\ c) `1 ) by MCART_1:7
.= ((a /\ b) `1 ) \/ ((a /\ c) `1 ) by MCART_1:7
.= ((a /\ b) \/ (a /\ c)) `1 by MCART_1:7 ; :: thesis: (a /\ (b \/ c)) `2 = ((a /\ b) \/ (a /\ c)) `2
thus (a /\ (b \/ c)) `2 = (a `2 ) /\ ((b \/ c) `2 ) by MCART_1:7
.= (a `2 ) /\ ((b `2 ) \/ (c `2 )) by MCART_1:7
.= ((a `2 ) /\ (b `2 )) \/ ((a `2 ) /\ (c `2 )) by XBOOLE_1:23
.= ((a `2 ) /\ (b `2 )) \/ ((a /\ c) `2 ) by MCART_1:7
.= ((a /\ b) `2 ) \/ ((a /\ c) `2 ) by MCART_1:7
.= ((a /\ b) \/ (a /\ c)) `2 by MCART_1:7 ; :: thesis: verum

end;

hence a /\ (b \/ c) = (a /\ b) \/ (a /\ c) by DOMAIN_1:12; :: thesis: verum