let a, b, c be Element of L; :: according to BINOP_1:def 3 :: thesis: the L_join of L . (a,( the L_join of L . (b,c))) = the L_join of L . (( the L_join of L . (a,b)),c)
thus the L_join of L . (a,( the L_join of L . (b,c))) = a "\/" (b "\/" c)
.= (a "\/" b) "\/" c by LATTICES:def 5
.= the L_join of L . (( the L_join of L . (a,b)),c) ; :: thesis: verum