
theorem
  for M being non empty MetrSpace, P, Q being non empty Subset of
TopSpaceMetr M, X being Subset of REAL st X = (dist_max P) .: Q & P is compact
  & Q is compact holds X is bounded_above
proof
  let M be non empty MetrSpace, P, Q be non empty Subset of TopSpaceMetr M, X
  be Subset of REAL;
  assume that
A1: X = (dist_max P).:Q and
A2: P is compact & Q is compact;
  reconsider Q9 = Q as non empty Subset of M by TOPMETR:12;
  X is bounded_above
  proof
    take r = max_dist_max (P, Q);
    let p be ExtReal;
    assume p in X;
    then consider z being object such that
    z in dom dist_max P and
A3: z in Q and
A4: p = (dist_max P).z by A1,FUNCT_1:def 6;
    z in Q9 by A3;
    then reconsider z as Point of M;
    (dist_max P) . z <= r by A2,A3,Th23;
    hence thesis by A4;
  end;
  hence thesis;
end;
