let A, B be non empty preBoolean set ; :: thesis: for a, b, c being Element of [:A,B:] holds (a /\ b) /\ c = a /\ (b /\ c)
let a, b, c be Element of [:A,B:]; :: thesis: (a /\ b) /\ c = a /\ (b /\ c)
A1: ((a /\ b) /\ c) `2 = ((a /\ b) `2) /\ (c `2)
.= ((a `2) /\ (b `2)) /\ (c `2)
.= (a `2) /\ ((b `2) /\ (c `2)) by XBOOLE_1:16
.= (a `2) /\ ((b /\ c) `2)
.= (a /\ (b /\ c)) `2 ;
((a /\ b) /\ c) `1 = ((a /\ b) `1) /\ (c `1)
.= ((a `1) /\ (b `1)) /\ (c `1)
.= (a `1) /\ ((b `1) /\ (c `1)) by XBOOLE_1:16
.= (a `1) /\ ((b /\ c) `1)
.= (a /\ (b /\ c)) `1 ;
hence (a /\ b) /\ c = a /\ (b /\ c) by A1, DOMAIN_1:2; :: thesis: verum