reserve A for set,
  C for non empty set,
  B for Subset of A,
  x for Element of A,
  f,g for Function of A,C;
reserve B for Element of Fin A;
reserve L for non empty LattStr,
  a,b,c for Element of L;
reserve L for Lattice;
reserve a,b,c,u,v for Element of L;
reserve A for non empty set,
  x for Element of A,
  B for Element of Fin A,
  f,g for Function of A, the carrier of L;
reserve L for 0_Lattice,
  f,g for Function of A, the carrier of L,
  u for Element of L;
reserve L for 1_Lattice,
  f,g for Function of A, the carrier of L,
  u for Element of L;
reserve L for D0_Lattice,
  f,g for (Function of A, the carrier of L),
  u for Element of L;

theorem Th64:
  (the L_meet of L).(u, FinJoin(B, f)) = FinJoin(B, (the L_meet of L)[;](u,f))
proof
A1: (the L_meet of L).(u,Bottom L) = u"/\"Bottom L .= Bottom L;
  the L_meet of L is_distributive_wrt the L_join of L & Bottom L =
  the_unity_wrt the L_join of L by Th18,Th23;
  hence thesis by A1,SETWOP_2:12;
end;
