
theorem
  for X be non empty MetrSpace, Y be SetSequence of X st X is complete &
  union rng Y = the carrier of X & for n be Nat holds (Y.n)` in
Family_open_set X holds ex n0 be Nat,
    r be Real, x0 be Point of X st
  0 < r & Ball(x0,r) c= Y.n0
proof
  let X be non empty MetrSpace, Y be SetSequence of X;
  assume that
A1: X is complete and
A2: union rng Y = the carrier of X and
A3: for n be Nat holds (Y.n)` in Family_open_set X;
  defpred P[Nat,Point of X,Real,Point of X,Real] means ( 0 < $3 &
$3 < 1/(2|^$1) & Ball($2,$3) /\ Y.($1) = {}) implies ( 0 < $5 & $5 < 1/(2|^($1+
  1)) & Ball($4,$5) c= Ball($2,($3)/2) & Ball($4,$5) /\ Y.($1+1) = {} );
  assume
A4: not ex n0 be Nat, r be Real,x0 be Point of X st 0 < r &
  Ball(x0,r) c= Y.n0;
  now
    set x0 = the Point of X;
A5: (Y.0)`` = (the carrier of X) \ ((Y.0)`) & Ball(x0,1) c= the carrier of X;
    assume (Y.0)` = {};
    hence contradiction by A4,A5;
  end;
  then consider z0 be object such that
A6: z0 in (Y.0)` by XBOOLE_0:def 1;
  reconsider z0 as Element of X by A6;
  (Y.0)` in Family_open_set X by A3;
  then consider t01 be Real such that
A7: 0 < t01 and
A8: Ball(z0,t01) c= (Y.0)` by A6,PCOMPS_1:def 4;
  reconsider t0=min(t01,1/2) as Element of REAL by XREAL_0:def 1;
  t0 <= 1/2 by XXREAL_0:17;
  then t0 < 1/1 by XXREAL_0:2;
  then
A9: t0 < 1/(2 |^0) by NEWTON:4;
  Ball(z0,t0) c= Ball(z0,t01) by PCOMPS_1:1,XXREAL_0:17;
  then Ball(z0,t0) c= (Y.0)` by A8;
  then Ball(z0,t0) misses (Y.0) by SUBSET_1:23;
  then
A10: Ball(z0,t0) /\ Y.0 ={} by XBOOLE_0:def 7;
A11: for n being Nat, x be Point of X, r be Real ex x1 be Point
of X,r1 be Real st 0 < r & r < 1/(2|^n) & Ball(x,r) /\ Y.n = {} implies 0 < r1
& r1 < 1/(2|^(n+1)) & Ball(x1,r1) c= Ball(x,r/2) & Ball(x1,r1) /\ Y.(n+1) = {}
  proof
    let n be Nat, x be Point of X, r be Real;
    now
      0 < 2|^(n+2) by NEWTON:83;
      then
A12:  0 < 1/(2|^(n+2)) by XREAL_1:139;
      0 < 2|^(n+1) by NEWTON:83;
      then
A13:  1/(2|^(n+1))/2 < 1/(2|^(n+1)) by XREAL_1:139,216;
      2|^(n+2) = 2|^((n+1)+1) .= 2|^(n+1)*2 by NEWTON:6;
      then
A14:  1/(2|^(n+2)) = 1/(2|^(n+1))/2 by XCMPLX_1:78;
      assume that
A15:  0 < r and
      r < 1/(2|^n) and
      Ball(x,r) /\ Y.n ={};
      not Ball(x,r/2) c= Y.(n+1) by A4,A15,XREAL_1:215;
      then Ball(x,r/2) meets (Y.(n+1))` by SUBSET_1:24;
      then consider z0 be object such that
A16:  z0 in Ball(x,r/2) /\ (Y.(n+1))` by XBOOLE_0:4;
      reconsider x1=z0 as Point of X by A16;
A17:  (Y.(n+1))` in Family_open_set X by A3;
      Ball(x,r/2) in Family_open_set X & (Y.(n+1))` in Family_open_set X
      by A3,PCOMPS_1:29;
      then Ball(x,r/2) /\ (Y.(n+1))` in Family_open_set X by PCOMPS_1:31;
      then consider t02 be Real such that
A18:  0 < t02 and
A19:  Ball(x1,t02) c= Ball(x,r/2) /\ (Y.(n+1))` by A16,PCOMPS_1:def 4;
A20:  Ball(x1,t02) c= Ball(x,r/2) by A19,XBOOLE_1:18;
      x1 in (Y.(n+1))` by A16,XBOOLE_0:def 4;
      then consider t01 be Real such that
A21:  0 < t01 and
A22:  Ball(x1,t01) c= (Y.(n+1))` by A17,PCOMPS_1:def 4;
      reconsider r1=min(min(t01,t02),1/(2|^(n+2))) as Real;
A23:  r1 <= min(t01,t02) by XXREAL_0:17;
      min(t01,t02) <= t02 by XXREAL_0:17;
      then
A24:  Ball(x1,r1) c= Ball(x1,t02) by A23,PCOMPS_1:1,XXREAL_0:2;
      min(t01,t02) <= t01 by XXREAL_0:17;
      then Ball(x1,r1) c= Ball(x1,t01) by A23,PCOMPS_1:1,XXREAL_0:2;
      then Ball(x1,r1) c= (Y.(n+1))` by A22;
      then
A25:  Ball(x1,r1) misses (Y.(n+1)) by SUBSET_1:23;
      take x1,r1;
A26:  r1 <= 1/(2|^(n+2)) by XXREAL_0:17;
      0 < min(t01,t02) by A21,A18,XXREAL_0:15;
      hence P[n,x,r,x1,r1] by A20,A14,A12,A13,A24,A25,A26,XBOOLE_0:def 7
,XBOOLE_1:1,XXREAL_0:2,15;
    end;
    hence thesis;
  end;
  ex x0 be sequence of X, r0 be Real_Sequence st x0.0=z0 & r0.0=t0 & for
n be Nat holds ( 0 < r0.n & r0.n < 1/(2|^n) & Ball(x0.n,r0.n) /\ Y.n
  = {} implies 0 < r0.(n+1) & r0.(n+1) < 1/(2|^(n+1)) & Ball(x0.(n+1),r0.(n+1))
  c= Ball(x0.n,(r0.n)/2) & Ball(x0.(n+1),r0.(n+1)) /\ Y.(n+1) = {} )
  proof
    defpred P1[Nat,Element of [:the carrier of X,REAL:], Element of
[:the carrier of X,REAL:]] means ( 0 < $2`2 & $2`2 < 1/(2|^$1) & Ball($2`1,$2`2
) /\ Y.($1) = {}) implies ( 0 < $3`2 & $3`2 < 1/(2|^($1+1)) & Ball($3`1,$3`2)
    c= Ball($2`1,($2`2)/2) & Ball($3`1,$3`2) /\ Y.($1+1) = {} );
A27: for n being Nat for u being Element of [:the carrier of X,
    REAL:] ex v being Element of [:the carrier of X,REAL:] st P1[n,u,v]
    proof
      let n be Nat, u be Element of [:the carrier of X,REAL:];
      consider v1 being Element of X, v2 being Real such that
A28:  P[n,u`1,u`2,v1,v2] by A11;
       reconsider v2 as Element of REAL by XREAL_0:def 1;
      take [v1,v2];
      thus thesis by A28;
    end;
    consider f being sequence of [:the carrier of X,REAL:] such that
A29: f.0 = [z0,t0] & for n being Nat holds P1[n,f.n,f.(n+1)
    ] from RECDEF_1:sch 2(A27);
    take pr1 f, pr2 f;
    thus (pr1 f).0 = (f.0)`1 by FUNCT_2:def 5
      .= z0 by A29;
    thus (pr2 f).0 = (f.0)`2 by FUNCT_2:def 6
      .= t0 by A29;
    hereby
      let i be Nat;
A30:    i in NAT by ORDINAL1:def 12;
A31:  (f.(i+1))`1 = (pr1 f).(i+1) & (f.(i+1))`2 = (pr2 f).(i+1) by
FUNCT_2:def 5,def 6;
      (f.i)`1 = (pr1 f).i & (f.i)`2 = (pr2 f).i by FUNCT_2:def 5,def 6,A30;
      hence P[i,(pr1 f).i,(pr2 f).i,(pr1 f).(i+1),(pr2 f).(i+1)] by A29,A31;
    end;
  end;
  then consider x0 be sequence of X, r0 be Real_Sequence such that
A32: x0.0=z0 & r0.0=t0 and
A33: for n be Nat holds ( 0 < r0.n & r0.n < 1/(2|^n) & Ball(
x0.n,r0.n) /\ Y.n = {} implies 0 < r0.(n+1) & r0.(n+1) < 1/(2|^(n+1)) & Ball (
x0.(n+1),r0.(n+1)) c= Ball(x0.n,(r0.n)/2) & Ball(x0.(n+1),r0.(n+1)) /\ Y.(n+1)
  = {} );
  0 < 1/2;
  then
A34: 0 < t0 by A7,XXREAL_0:15;
A35: for n be Nat holds 0 < r0.n & r0.n < 1/(2|^n) & Ball(x0.(n+1
  ),r0.(n+1)) c= Ball(x0.n,(r0.n)/2) & Ball(x0.n,r0.n) /\ Y.n ={}
  proof
    defpred PN[Nat] means
    0 < r0.$1 & r0.$1 < 1/(2|^$1) & Ball(x0.(
    $1+1),r0.($1+1)) c= Ball(x0.$1,(r0.$1)/2) & Ball(x0.$1,r0.$1) /\ Y.$1 = {};
A36: now
      let n be Nat;
      assume
A37:  PN[n];
      then
A38:  Ball(x0.(n+1),r0.(n+1)) /\ Y.(n+1) = {} by A33;
      0 < r0.(n+1) & r0.(n+1) < 1/(2|^(n+1)) by A33,A37;
      hence PN[n+1] by A33,A38;
    end;
A39: PN[0] by A34,A9,A10,A32,A33;
    thus for n be Nat holds PN[n] from NAT_1:sch 2(A39,A36);
  end;
A40: for m,k be Nat holds dist(x0.(m+k),x0.m) <= 1/(2|^m)*(1-1/(2
  |^k))
  proof
    let m be Nat;
    defpred PN[Nat] means
    dist(x0.(m+$1),x0.m) <= 1/(2|^m)*(1-1/(2|^$1));
A41: now
      let k be Nat;
      assume PN[k];
      then
A42:  dist(x0.((m+k)+1),x0.(m+k)) + dist(x0.(m+k),x0.m) <= dist(x0.((m+k)
      +1),x0.(m+k)) + 1/(2|^m)*(1-1/(2|^k)) by XREAL_1:6;
      0 < r0.(m+k+1) & dist(x0.(m+k+1),x0.(m+k+1)) = 0 by A35,METRIC_1:1;
      then
A43:  x0.(m+k+1) in Ball(x0.(m+k+1),r0.(m+k+1)) by METRIC_1:11;
      r0.(m+k) < 1/(2|^(m+k)) by A35;
      then r0.(m+k)/2 < 1/(2|^(m+k))/2 by XREAL_1:74;
      then r0.(m+k)/2 < 1/(2|^(m+k)*2) by XCMPLX_1:78;
      then
A44:  r0.(m+k)/2 < 1/(2|^(m+k+1)) by NEWTON:6;
      Ball(x0.(m+k+1),r0.(m+k+1)) c= Ball(x0.(m+k),r0.(m+k)/2) by A35;
      then dist(x0.(m+k+1),x0.(m+k)) < r0.(m+k)/2 by A43,METRIC_1:11;
      then dist(x0.(m+k+1),x0.(m+k)) <= 1/(2|^(m+k+1)) by A44,XXREAL_0:2;
      then
A45:  dist(x0.((m+k)+1),x0.(m+k)) + 1/(2|^m)*(1-1/(2|^k)) <= 1/(2|^(m+k+1
      )) + 1/(2|^m)*(1-1/(2|^k)) by XREAL_1:6;
      2|^(m+(k+1)) = 2|^m * 2|^(k+1) by NEWTON:8;
      then 1/(2|^(m+k+1)) = 1/(2|^m)/((2|^(k+1))/1) by XCMPLX_1:78
        .= 1/(2|^m)*(1/(2|^(k+1))) by XCMPLX_1:79;
      then
A46:  1/(2|^(m+k+1)) + 1/(2|^m)*(1-1/(2|^k)) = 1/(2|^m)*( 1 + ( 1/(2|^(k+
      1)) - 1/(2|^k) ) )
        .= 1/(2|^m)*( 1 + ( 1/((2|^k)*2) - 1/(2|^k) ) ) by NEWTON:6
        .= 1/(2|^m)*( 1 + ( 1/(2|^k)/2 - 1/(2|^k) ) ) by XCMPLX_1:78
        .= 1/(2|^m)*( 1 - 1/(2|^k)/2 )
        .= 1/(2|^m)*( 1 - 1/((2|^k)*2) ) by XCMPLX_1:78;
      dist(x0.((m+k)+1),x0.m) <= dist(x0.((m+k)+1),x0.(m+k)) + dist(x0.(m
      +k),x0.m) by METRIC_1:4;
      then
      dist(x0.(m+(k+1)),x0.m) <= dist(x0.((m+k)+1),x0.(m+k)) + 1/(2|^m)*(
      1-1/(2|^k)) by A42,XXREAL_0:2;
      then dist(x0.(m+(k+1)),x0.m) <= 1/(2|^(m+k+1) ) + 1/(2|^m)*(1-1/(2|^k))
      by A45,XXREAL_0:2;
      hence PN[k+1] by A46,NEWTON:6;
    end;
    2|^0 = 1 by NEWTON:4;
    then
A47: PN[0] by METRIC_1:1;
    for k be Nat holds PN[k] from NAT_1:sch 2(A47,A41);
    hence thesis;
  end;
A48: now
    let m be Nat;
    hereby
      let k be Nat;
A49:  dist(x0.(m+k),x0.m) <= 1/(2|^m)*(1-1/(2|^k)) by A40;
      0 < 2|^k by NEWTON:83;
      then 0 < 1/(2|^k) by XREAL_1:139;
      then
A50:  1-1/(2|^k) < 1-0 by XREAL_1:10;
      0 < 2|^m by NEWTON:83;
      then 0 < 1/(2|^m) by XREAL_1:139;
      then 1/(2|^m)*(1-1/(2|^k)) < 1/(2|^m)*1 by A50,XREAL_1:68;
      hence dist(x0.(m+k),x0.m) < 1/(2|^m) by A49,XXREAL_0:2;
    end;
  end;
  now
    let r be Real;
    assume 0 < r;
    then consider p1 be Nat such that
A51: 1/(2|^p1) <= r by PREPOWER:92;
     reconsider p=p1+1 as Nat;
    take p;
    hereby
      let n,m be Nat;
      assume that
A52:  n >= p and
A53:  m >= p;
      consider k1 be Nat such that
A54:  n = p + k1 by A52,NAT_1:10;
      reconsider k1 as Nat;
      n = p + k1 by A54;
      then
A55:  dist(x0.n,x0.p) < 1/(2|^p) by A48;
      consider k2 be Nat such that
A56:  m = p + k2 by A53,NAT_1:10;
      reconsider k2 as Nat;
A57:  1/(2|^p) = 1/((2|^p1)*2) by NEWTON:6
        .= 1/(2|^p1)/2 by XCMPLX_1:78;
      m = p + k2 by A56;
      then
A58:  dist(x0.n,x0.p) + dist(x0.p,x0.m) < 1/(2|^p) + 1/(2|^p) by A48,A55,
XREAL_1:8;
      dist(x0.n,x0.m) <= dist(x0.n,x0.p) + dist(x0.p,x0.m) by METRIC_1:4;
      then dist(x0.n,x0.m) < 1/(2|^p1) by A58,A57,XXREAL_0:2;
      hence dist(x0.n,x0.m)<r by A51,XXREAL_0:2;
    end;
  end;
  then x0 is Cauchy by TBSP_1:def 4;
  then
A59: x0 is convergent by A1,TBSP_1:def 5;
A60: for m,k be Nat holds Ball(x0.(m+1+k),r0.(m+1+k)) c= Ball(x0
  .m,(r0.m)/2)
  proof
    let m be Nat;
    defpred PN[Nat] means
    Ball(x0.(m+1+$1),r0.(m+1+$1)) c= Ball(x0.m,(r0.m)/2);
A61: now
      let k be Nat;
      assume
A62:  PN[k];
      0 < r0.(m+1+k) by A35;
      then (r0.(m+1+k))/2 < r0.(m+1+k) by XREAL_1:216;
      then
A63:  Ball(x0.(m+1+k),(r0.(m+1+k))/2) c=Ball(x0.(m+1+k),r0.(m+1+k)) by
PCOMPS_1:1;
      Ball(x0.(m+1+(k+1)),r0.(m+1+(k+1)))=Ball(x0.(m+1+k+1),r0.(m+1+k+1) );
      then Ball(x0.(m+1+(k+1)),r0.(m+1+(k+1))) c= Ball(x0.(m+1+k),(r0.(m+ 1+k
      ))/ 2) by A35;
      then Ball(x0.(m+1+(k+1)),r0.(m+1+(k+1))) c= Ball(x0.(m+1+k),r0.(m+1+k))
      by A63;
      hence PN[k+1] by A62,XBOOLE_1:1;
    end;
A64: PN[0] by A35;
    thus for k be Nat holds PN[k] from NAT_1:sch 2(A64,A61);
  end;
A65: now
    let m be Nat;
    set m1 = m+1;
    0 < r0.m by A35;
    then 0 < (r0.m)/2 by XREAL_1:215;
    then consider n1 be Nat such that
A66: for l be Nat st n1 <= l holds dist(x0.l,lim x0) < (r0
    .m)/2 by A59,TBSP_1:def 3;
    reconsider n = max(n1,m+1) as Nat by TARSKI:1;
A67: dist(x0.n,lim x0) < (r0.m)/2 by A66,XXREAL_0:25;
    consider k be Nat such that
A68: n = m1 + k by NAT_1:10,XXREAL_0:25;
    dist(x0.n,x0.n) = 0 & 0 < r0.n by A35,METRIC_1:1;
    then
A69: x0.n in Ball(x0.n,r0.n) by METRIC_1:11;
    reconsider k as Nat;
    n = m1 + k by A68;
    then Ball(x0.n,r0.n) c= Ball(x0.m,(r0.m)/2) by A60;
    then dist(x0.n,x0.m) < (r0.m)/2 by A69,METRIC_1:11;
    then
A70: dist(lim x0,x0.n) + dist(x0.n,x0.m) < (r0.m)/2 + (r0.m)/2 by A67,XREAL_1:8
;
    dist(lim x0,x0.m) <= dist(lim x0,x0.n) +dist(x0.n,x0.m) by METRIC_1:4;
    then dist(lim x0,x0.m) < (r0.m)/2 + (r0.m)/2 by A70,XXREAL_0:2;
    hence lim x0 in Ball(x0.m,r0.m) by METRIC_1:11;
  end;
A71: now
    let n be Nat;
    thus not lim x0 in Y.n
    proof
      assume
A72:  lim x0 in Y.n;
      lim x0 in Ball(x0.n,r0.n) by A65;
      then lim x0 in Ball(x0.n,r0.n) /\ Y.n by A72,XBOOLE_0:def 4;
      hence contradiction by A35;
    end;
  end;
  not lim x0 in union rng Y
  proof
    assume lim x0 in union rng Y;
    then consider A be set such that
A73: lim x0 in A and
A74: A in rng Y by TARSKI:def 4;
    ex k be object st k in dom Y & A = Y.k by A74,FUNCT_1:def 3;
    hence contradiction by A71,A73;
  end;
  hence contradiction by A2;
end;
