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;

theorem Th59:
  (for x st x in B holds u [= f.x) implies u [= FinMeet(B,f)
proof
  reconsider f9 = f as Function of A, the carrier of L.:;
  reconsider u9 = u as Element of L.:;
  assume for x st x in B holds u [= f.x;
  then
A1: for x st x in B holds f9.x [= u9 by Th38;
  L.: is 0_Lattice by Th49;
  then FinJoin(B,f9) [= u9 by A1,Th54;
  hence thesis by Th39;
end;
