let 0L be lower-bounded Lattice; :: thesis: for B1, B2 being Element of Fin the carrier of 0L holds (FinJoin B1) "\/" (FinJoin B2) = FinJoin (B1 \/ B2)
let B1, B2 be Element of Fin the carrier of 0L; :: thesis: (FinJoin B1) "\/" (FinJoin B2) = FinJoin (B1 \/ B2)
set J = the L_join of 0L;
thus FinJoin (B1 \/ B2) = the L_join of 0L $$ ((B1 \/ B2),(id 0L)) by LATTICE2:def 3
.= ( the L_join of 0L $$ (B1,(id 0L))) "\/" ( the L_join of 0L $$ (B2,(id 0L))) by SETWISEO:33
.= (FinJoin B1) "\/" ( the L_join of 0L $$ (B2,(id 0L))) by LATTICE2:def 3
.= (FinJoin B1) "\/" (FinJoin B2) by LATTICE2:def 3 ; :: thesis: verum