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;
reserve DL for distributive upper-bounded Lattice,
  B for Element of Fin the carrier of DL,
  p for Element of DL,
  f for UnOp of the carrier of DL;

theorem Th25:
  FinMeet B "\/" p = FinMeet (((the L_join of DL)[:](id DL,p)).:B)
proof
  set J = the L_join of DL;
  set M = the L_meet of DL;
  thus FinMeet B "\/" p = J.(M $$ (B,id DL),p) by LATTICE2:def 4
    .= M $$ (B,J [:] (id DL,p)) by Lm3
    .=FinMeet(B,J [:] (id DL,p)) by LATTICE2:def 4
    .= FinMeet (B,(id DL)*(J [:] (id DL,p))) by FUNCT_2:17
    .= FinMeet (((the L_join of DL)[:](id DL,p)).: B) by Th22;
end;
