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

theorem
  TopSpaceMetr M is compact & F is Cover of TopSpaceMetr M & F c= G
    implies L is Lebesgue_number of G
  proof
  assume that
   A1: TopSpaceMetr M is compact and
   A2: F is Cover of TopSpaceMetr M and
   A3: F c=G;
  A4: now let x be Point of M;
   ex A be Subset of TopSpaceMetr M st A in F & Ball(x,L)c=A by A1,A2,Def1;
   hence ex A be Subset of TopSpaceMetr M st A in G & Ball(x,L)c=A by A3;
  end;
  set TM=TopSpaceMetr M;
  union F=[#]TM & union F c=union G by A2,A3,SETFAM_1:45,ZFMISC_1:77;
  then G is Cover of TM by SETFAM_1:def 11;
  hence thesis by A1,A4,Def1;
 end;
