 reserve x,X for set,
         n, m, i for Nat,
         p, q for Point of TOP-REAL n,
         A, B for Subset of TOP-REAL n,
         r, s for Real;
reserve N for non zero Nat,
        u,t for Point of TOP-REAL(N+1);

theorem Th13:
  for r st r>0
    for U be Subset of Tdisk(p,r) st U is open non empty
    for A be Subset of TOP-REAL n st A = U holds Int A is non empty
proof
  set TR=TOP-REAL n;
  let r be Real such that
A1: r>0;
  let U be Subset of Tdisk(p,r) such that
A2: U is open non empty;
  consider q be object such that
A3: q in U by A2;
A4: [#]Tdisk(p,r) = cl_Ball(p,r) by PRE_TOPC:def 5;
  then q in cl_Ball(p,r) by A3;
  then reconsider q as Point of TR;
  the TopStruct of TR=TopSpaceMetr Euclid n by EUCLID:def 8;
  then reconsider Q=q as Point of Euclid n by TOPMETR:12;
A5: |.q-p.|<= r by A3,A4,TOPREAL9:8;
  U in the topology of Tdisk(p,r) by A2,PRE_TOPC:def 2;
  then consider W be Subset of TR such that
A6:   W in the topology of TR
    and
A7:   U = W /\ [#]Tdisk(p,r) by PRE_TOPC:def 4;
  reconsider W as open Subset of TR by A6,PRE_TOPC:def 2;
  Int W=W by TOPS_1:23;
  then q in Int (W) by A3,A7,XBOOLE_0:def 4;
  then consider s be Real such that
A8:   s>0
    and
A9:   Ball(Q,s) c= W by GOBOARD6:5;
  set m=min(s,r),mr=m/(2*r);
A10: 0< m by A8,A1, XXREAL_0:21;
  then
A11: m/2 < m by XREAL_1:216;
  set qp = (-mr)*(q-p)+q;
A12: qp -q = (-mr)*(q-p)+(q-q) by RLVECT_1:def 3
          .=(- mr)*(q-p)+0.TR by RLVECT_1:def 10
          .=(-mr)*(q-p) by RLVECT_1:def 4;
  |.-mr.| =--mr by A10, A1,ABSVALUE:def 1;
  then
A13: |.qp-q.| = mr *|.q-p.| by A12,EUCLID:11;
  mr *r = m/2/r*r by XCMPLX_1:78
       .= m/2*(r/r)
       .= m/2*1 by XCMPLX_1:60,A1
       .= m/2;
  then |.qp-q.| <= m/2 by A5,XREAL_1:66, A10,A13;
  then
A14: |.qp-q.| < m by A11,XXREAL_0:2;
  m <= s by XXREAL_0: 17;
  then |.qp-q.| <s by A14,XXREAL_0:2;
  then
A15: qp in Ball(q,s);
  let A be Subset of TR such that
A16: A=U;
A17: Ball(p,r) c= cl_Ball(p,r) by TOPREAL9:16;
A18: -mr+1 <0+1 by XREAL_1:8, A10, A1;
A19: (1-mr)*|. q-p .| < r
  proof
    per cases;
      suppose |. q-p .| =0;
        hence thesis by A1;
      end;
      suppose |. q-p .| >0;
        then (1-mr)*|. q-p .| < 1*|. q-p .| by A18,XREAL_1:68;
        hence thesis by A5,XXREAL_0:2;
      end;
  end;
A20: r/(2*r) = r/r/2 by XCMPLX_1:78
            .= 1/2 by XCMPLX_1:60,A1;
  mr <= r/(2*r) by A1, XXREAL_0:17, XREAL_1:72;
  then 1-mr >= 1-1/2 by A20,XREAL_1:10;
  then
A21: |.1-mr.|=1-mr by ABSVALUE:def 1;
  qp - p =(-mr)*(q-p)+(q-p) by RLVECT_1:def 3
        .=(-mr)*(q-p)+1*(q-p) by RLVECT_1:def 8
        .=(1-mr)*(q-p) by RLVECT_1:def 6;
  then |.qp - p.| <r by A19, A21,EUCLID:11;
  then qp in Ball(p,r);
  then
A22:qp in Ball(q,s)/\Ball(p,r) by A15,XBOOLE_0:def 4;
  Ball(q,s) c= W by A9, TOPREAL9:13;
  hence thesis by A17,XBOOLE_1:27,A7,A4,A16,A22,TOPS_1:22;
end;
