
theorem Th2:
  for M being MetrSpace, n being Nat, F being Subset-Family of M, p
  being FinSequence st F is being_ball-family & rng p = F & dom p = Seg(n+1)
holds ex G being Subset-Family of M st (G is finite & G is being_ball-family &
ex q being FinSequence st rng q = G & dom q = Seg(n) & ex x being Point of M st
  ex r being Real st union F c= union G \/ Ball(x,r) )
proof
  let M be MetrSpace;
  let n be Nat;
  let F be Subset-Family of M;
  let p be FinSequence;
  assume that
A1: F is being_ball-family and
A2: rng p = F and
A3: dom p = Seg(n+1);
  n+1 in dom p by A3,FINSEQ_1:4;
  then p.(n+1) in F by A2,FUNCT_1:def 3;
  then consider x being Point of M such that
A4: ex r being Real st p.(n+1) = Ball(x,r) by A1,TOPMETR:def 4;
  consider r being Real such that
A5: p.(n+1) = Ball(x,r) by A4;
  reconsider q = p|(Seg(n)) as FinSequence by FINSEQ_1:15;
A6: rng q c= rng p by RELAT_1:70;
  then reconsider G=rng q as Subset-Family of M by A2,XBOOLE_1:1;
  reconsider G as Subset-Family of M;
  len p = n+1 by A3,FINSEQ_1:def 3;
  then n <= len p by NAT_1:11;
  then
A7: dom q = Seg(n) by FINSEQ_1:17;
  then
A8: dom q \/ {n+1} = dom p by A3,FINSEQ_1:9;
A9: ex x being Point of M st
   ex r being Real st union F c= union G \/ Ball(x,r)
  proof
    take x;
     reconsider r as Real;
    take r;
    union F c= union G \/ Ball(x,r)
    proof
      let t be object;
      assume t in union F;
      then consider A being set such that
A10:  t in A and
A11:  A in F by TARSKI:def 4;
      consider s being object such that
A12:  s in dom p and
A13:  A = p.s by A2,A11,FUNCT_1:def 3;
      now
        per cases by A8,A12,XBOOLE_0:def 3;
        case
          s in dom q;
          then q.s in G & q.s = A by A13,FUNCT_1:47,def 3;
          then
A14:      t in union G by A10,TARSKI:def 4;
          union G c= union G \/ Ball(x,r) by XBOOLE_1:7;
          hence thesis by A14;
        end;
        case
          s in {n+1};
          then p.s = Ball(x,r) by A5,TARSKI:def 1;
          hence thesis by A10,A13,XBOOLE_0:def 3;
        end;
      end;
      hence thesis;
    end;
    hence thesis;
  end;
  take G;
  for x being set holds x in G implies ex y being Point of M st
   ex r being Real st x = Ball(y,r) by A1,A2,A6,TOPMETR:def 4;
  hence thesis by A7,A9,TOPMETR:def 4;
end;
