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;

theorem Th41:
  (ex x st x in B & f.x [= u) implies FinMeet(B,f)[= u
proof
  given x such that
A1: x in B and
A2: f.x [= u;
  reconsider u9 = u as Element of L.:;
  reconsider f9 = f as Function of A, the carrier of L.:;
  u9 [= f9.x by A2,Th38;
  then u9 [= FinJoin(B,f9) by A1,Th29;
  hence thesis by Th39;
end;
