let a, b, c be Element of L; :: according to BINOP_1:def 14 :: 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