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
  B <> {} & (for x st x in B holds f.x [= g.x) implies FinJoin(B,f) [=
  FinJoin(B,g)
proof
  assume that
A1: B <> {} and
A2: for x st x in B holds f.x [= g.x;
  now
    let x;
    assume
A3: x in B;
    then f.x [= g.x by A2;
    hence f.x [= FinJoin(B,g) by A3,Th29;
  end;
  hence thesis by A1,Th32;
end;
