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 is_finer_than 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 is_finer_than G;
  set TM=TopSpaceMetr M;
  A4: now let x be Point of M;
   consider A be Subset of TopSpaceMetr M such that
    A5: A in F and
    A6: Ball(x,L)c=A by A1,A2,Def1;
   consider B be set such that
    A7: B in G and
    A8: A c=B by A3,A5,SETFAM_1:def 2;
   reconsider B as Subset of TM by A7;
   take B;
   thus B in G & Ball(x,L)c=B by A6,A7,A8;
  end;
  union F=[#]TM & union F c=union G by A2,A3,SETFAM_1:13,45;
  then G is Cover of TM by SETFAM_1:def 11;
  hence thesis by A1,A4,Def1;
 end;
