
theorem Th3:
  for M being MetrSpace, F being Subset-Family of M st F is finite
  & F is being_ball-family holds ex x being Point of M, r being Real st union F
  c= Ball(x,r)
proof
  let M be MetrSpace;
  let F be Subset-Family of M;
  assume that
A1: F is finite and
A2: F is being_ball-family;
  consider p being FinSequence such that
A3: rng p = F by A1,FINSEQ_1:52;
A4: for F being Subset-Family of M st F is finite & F is being_ball-family
holds for n being Nat holds for p being FinSequence st rng p = F & dom p = Seg
  n holds ex x being Point of M, r being Real st union F c= Ball(x,r)
  proof
    defpred P[Nat] means for F being Subset-Family of M st (F is finite & F is
    being_ball-family) holds for n being Nat st n = $1 holds for p being
FinSequence st (rng p = F & dom p = Seg(n)) holds (ex x being Point of M st ex
    r being Real st union F c= Ball(x,r));
A5: for k being Nat st P[k] holds P[k + 1]
    proof
      let k be Nat;
      assume
A6:   P[k];
      let F be Subset-Family of M;
      assume that
      F is finite and
A7:   F is being_ball-family;
      let n be Nat;
      assume
A8:   n = k+1;
      let p be FinSequence;
      assume rng p = F & dom p = Seg(n);
      then consider F1 being Subset-Family of M such that
A9:   F1 is finite & F1 is being_ball-family and
A10:  ex p1 being FinSequence st rng p1 = F1 & dom p1 = Seg(k) & ex
x2 being Point of M st ex r2 being Real st union F c= union F1 \/ Ball(x2,r2)
      by A7,A8,Th2;
      consider x1 being Point of M such that
A11:  ex r being Real st union F1 c= Ball(x1,r) by A6,A9,A10;
      consider x2 being Point of M such that
A12:  ex r2 being Real st union F c= union F1 \/ Ball(x2,r2) by A10;
      consider r2 being Real such that
A13:  union F c= union F1 \/ Ball(x2,r2) by A12;
      consider r1 being Real such that
A14:  union F1 c= Ball(x1,r1) by A11;
      consider x being Point of M such that
A15:  ex r being Real st Ball(x1,r1) \/ Ball(x2,r2) c= Ball(x,r) by Th1;
      take x;
      consider r being Real such that
A16:  Ball(x1,r1) \/ Ball(x2,r2) c= Ball(x,r) by A15;
       reconsider r as Real;
      take r;
      union F1 \/ Ball(x2,r2) c= Ball(x1,r1) \/ Ball(x2,r2) by A14,XBOOLE_1:9;
      then union F c= Ball(x1,r1) \/ Ball(x2,r2) by A13;
      hence thesis by A16;
    end;
A17: P[0]
    proof
      let F be Subset-Family of M;
      assume that
      F is finite and
      F is being_ball-family;
      let n be Nat;
      assume n = 0;
      then
A18:  Seg n = {};
      for p being FinSequence st rng p = F & dom p = Seg(n) holds ex x
      being Point of M st ex r being Real st union F c= Ball(x,r)
      proof
        set x = the Point of M;
        let p be FinSequence;
        assume
A19:    rng p = F & dom p = Seg(n);
        take x, 0;
        union F = {} by A19,A18,RELAT_1:42,ZFMISC_1:2;
        hence thesis;
      end;
      hence thesis;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(A17,A5);
    hence thesis;
  end;
  ex n being Nat st dom p = Seg n by FINSEQ_1:def 2;
  hence thesis by A2,A4,A3;
end;
