reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem LMCLOSE2:
  for E be RealNormSpace,
      r be Real,
      z,y1,y2 be Point of E
    st y1 in cl_Ball(z,r) & y2 in cl_Ball(z,r)
  holds [.y1,y2.] c= cl_Ball(z,r)
  proof
    let E be RealNormSpace,
        r be Real,
        z,y1,y2 be Point of E;
    assume
    A1: y1 in cl_Ball(z,r) & y2 in cl_Ball(z,r); then
    A2: ex y1q be Element of E
        st y1 = y1q & ||.z - y1q .|| <= r;
    A3: ex y2q be Element of E
        st y2 = y2q & ||.z - y2q.|| <= r by A1;
    A4: [.y1,y2.]
      = { (((1 - r) * y1) + (r * y2)) where r is Real : 0 <= r <= 1 }
          by RLTOPSP1:def 2;
      let s be object;
      assume s in [.y1,y2.]; then
      consider p be Real such that
      A5: s = (1 - p) * y1 + (p * y2) & 0 <= p & p <= 1 by A4;
      reconsider w = s as Point of E by A5;
      (1 - p) * z + p * z = ((1-p)+p) * z by RLVECT_1:def 6
       .= z by RLVECT_1:def 8; then
      z - w
        = (1 - p) * z + p * z - (1 - p) * y1 - (p * y2) by A5,RLVECT_1:27
       .= (1 - p) * z + - (1 - p) * y1 + p * z - (p * y2) by RLVECT_1:def 3
       .= ((1 - p) * z  - (1 - p) * y1) +(p * z + - (p * y2)) by RLVECT_1:def 3
       .= (1 - p) * (z - y1) + (p * z  - p * y2) by RLVECT_1:34
       .= (1 - p) * (z - y1) + p * (z -y2) by RLVECT_1:34; then
      ||.z - w.|| <= ||. (1 - p) * (z  - y1) .|| + ||. p*(z -y2) .||
          by NORMSP_1:def 1; then
      ||.z - w.|| <= |. 1-p .| * ||. z  - y1 .|| + ||. p*(z -y2) .||
          by NORMSP_1:def 1; then
      A7: ||.z - w.|| <= |. 1-p .| * ||. z  - y1 .|| + |.p.| *||. z -y2 .||
          by NORMSP_1:def 1;
      A8: |. 1 - p .| = 1 - p & |.p.| = p by A5,COMPLEX1:43,XREAL_1:48;
      0 <= 1 - p by A5,XREAL_1:48; then
      A9: (1 - p) * ||. z  - y1 .|| <= (1 - p) * r by A2,XREAL_1:64;
      p * ||. z -y2 .|| <= p * r by A3,A5,XREAL_1:64; then
      (1 - p) * ||. z  - y1 .|| + p * ||. z -y2 .|| <= (1 - p) * r + p * r
          by A9,XREAL_1:7; then
      ||.z - w.|| <= (1 - p) * r + p * r by A7,A8,XXREAL_0:2;
      hence thesis;
  end;
