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;

theorem Th54:
  (for x st x in B holds f.x [= u) implies FinJoin(B,f) [= u
proof
  assume
A1: for x st x in B holds f.x [= u;
  set J = the L_join of L;
A2: Bottom L = the_unity_wrt J by Th18;
  now
    per cases;
    suppose
      B = {};
      then FinJoin(B,f) = J$$({}.A,f) .= Bottom L by A2,SETWISEO:31;
      hence thesis;
    end;
    suppose
      B <> {};
      hence thesis by A1,Th32;
    end;
  end;
  hence thesis;
end;
