reserve M for non empty MetrSpace,
        F,G for open Subset-Family of TopSpaceMetr M;
reserve L for Lebesgue_number of F;

theorem
  for L1 be positive Real st TopSpaceMetr M is compact &
                             F is Cover of TopSpaceMetr M & L1 <= L
    holds L1 is Lebesgue_number of F
 proof
  let L9 be positive Real such that
   A1: (TopSpaceMetr M is compact) & F is Cover of TopSpaceMetr M and
   A2: L9<=L;
  now let x be Point of M;
   consider A be Subset of TopSpaceMetr M such that
    A3: A in F & Ball(x,L)c=A by A1,Def1;
   take A;
   Ball(x,L9)c=Ball(x,L) by A2,PCOMPS_1:1;
   hence A in F & Ball(x,L9)c=A by A3;
  end;
  hence thesis by A1,Def1;
 end;
