let V, C be set ; :: thesis: for a, b, c being Element of (SubstLatt V,C) holds a "\/" (b "\/" c) = (a "\/" b) "\/" c
let a, b, c be Element of (SubstLatt V,C); :: thesis: a "\/" (b "\/" c) = (a "\/" b) "\/" c
reconsider a' = a, b' = b, c' = c as Element of SubstitutionSet V,C by Def4;
set G = SubstLatt V,C;
a "\/" (b "\/" c) = the L_join of (SubstLatt V,C) . a,(mi (b' \/ c')) by Def4
.= mi ((mi (b' \/ c')) \/ a') by Def4
.= mi (a' \/ (b' \/ c')) by Th13
.= mi ((a' \/ b') \/ c') by XBOOLE_1:4
.= mi ((mi (a' \/ b')) \/ c') by Th13
.= the L_join of (SubstLatt V,C) . (mi (a' \/ b')),c' by Def4
.= (a "\/" b) "\/" c by Def4 ;
hence a "\/" (b "\/" c) = (a "\/" b) "\/" c ; :: thesis: verum