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 Th46:
  B <> {} & (for x st x in B holds u [= f.x) implies u [= FinMeet( B,f)
proof
  assume that
A1: B <> {} and
A2: for x st x in B holds u [= f.x;
  reconsider u9 = u as Element of L.:;
  reconsider f9 = f as Function of A, the carrier of L.:;
  for x st x in B holds f9.x [= u9 by A2,Th38;
  then FinJoin(B,f9) [= u9 by A1,Th32;
  hence thesis by Th39;
end;
