reserve x,y,X,X1,Y,Z for set;
reserve L for Lattice;
reserve F,H for Filter of L;
reserve p,q,r for Element of L;
reserve L1, L2 for Lattice;
reserve a1,b1 for Element of L1;
reserve a2 for Element of L2;
reserve f for Homomorphism of L1,L2;
reserve B for Element of Fin the carrier of L;
reserve DL for distributive Lattice;
reserve f for Homomorphism of DL,L2;
reserve 0L for lower-bounded Lattice,
  B,B1,B2 for Element of Fin the carrier of 0L,
  b for Element of 0L;
reserve f for UnOp of the carrier of 0L;
reserve 1L for upper-bounded Lattice,
  B,B1,B2 for Element of Fin the carrier of 1L,
  b for Element of 1L;
reserve f,g for UnOp of the carrier of 1L;

theorem Th23:
  FinMeet B1 "/\" FinMeet B2 = FinMeet (B1 \/ B2)
proof
  set M = the L_meet of 1L;
  thus FinMeet (B1 \/ B2) = M $$ (B1 \/ B2,id 1L) by LATTICE2:def 4
    .= M $$ (B1,id 1L) "/\" M $$ (B2,id 1L) by SETWISEO:33
    .= FinMeet B1 "/\" M $$ (B2,id 1L) by LATTICE2:def 4
    .=FinMeet B1 "/\" FinMeet B2 by LATTICE2:def 4;
end;
