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 a9 = a, b9 = b, c9 = 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 (b9 \/ c9))) by Def4
.= mi ((mi (b9 \/ c9)) \/ a9) by Def4
.= mi (a9 \/ (b9 \/ c9)) by Th13
.= mi ((a9 \/ b9) \/ c9) by XBOOLE_1:4
.= mi ((mi (a9 \/ b9)) \/ c9) by Th13
.= the L_join of (SubstLatt (V,C)) . ((mi (a9 \/ b9)),c9) by Def4
.= (a "\/" b) "\/" c by Def4 ;
hence a "\/" (b "\/" c) = (a "\/" b) "\/" c ; :: thesis: verum