reserve n for Nat,
  a, b, r, w for Real,
  x, y, z for Point of TOP-REAL n,
  e for Point of Euclid n;

theorem Th22:
  a + b = 1 & |.a.| + |.b.| = 1 & b <> 0 & x in cl_Ball(z,r) & y
  in Ball(z,r) implies a * x + b * y in Ball(z,r)
proof
  assume that
A1: a + b = 1 and
A2: |.a.| + |.b.| = 1 and
A3: b <> 0 and
A4: x in cl_Ball(z,r) and
A5: y in Ball(z,r);
  |. y-z .| < r by A5,Th5;
  then
A6: |. z-y .| < r by TOPRNS_1:27;
  |. x-z .| <= r by A4,Th6;
  then 0 <= |.a.| & |. z-x .| <= r by COMPLEX1:46,TOPRNS_1:27;
  then
A7: |.a.|*|. z-x .| <= |.a.|*r by XREAL_1:64;
  0 < |.b.| by A3,COMPLEX1:47;
  then |.b.|*|. z-y .| < |.b.|*r by A6,XREAL_1:68;
  then |.a.|*|. z-x .| + |.b.|*|. z-y .| < |.a.|*r + |.b.|*r by A7,XREAL_1:8;
  then a is Real & |.a.|*|. z-x .| + |.b.|*|. z-y .| < (|.a.|+|.b.|)*r;
  then b is Real & |. a*(z-x) .| + |.b.|*|. z-y .| < 1 * r by A2,TOPRNS_1:7;
  then
A8: |. a*(z-x) .| + |. b*(z-y) .| < r by TOPRNS_1:7;
  |. a*z + b*z - (a*x + b*y) .| = |. a*z - (a*x + b*y) + b*z .| by
RLVECT_1:def 3
    .= |. a*z - a*x - b*y + b*z .| by RLVECT_1:27
    .= |. a*z - a*x + b*z - b*y .| by RLVECT_1:def 3
    .= |. a*z - a*x + (b*z - b*y) .| by RLVECT_1:def 3
    .= |. a*(z-x) + (b*z - b*y) .| by RLVECT_1:34
    .= |. a*(z-x) + b*(z-y) .| by RLVECT_1:34;
  then |. a*z + b*z - (a*x + b*y) .| <= |. a*(z-x) .| + |. b*(z-y) .| by
TOPRNS_1:29;
  then |. a*z + b*z - (a*x + b*y) .| < r by A8,XXREAL_0:2;
  then
A9: |. a*x + b*y - (a*z + b*z) .| < r by TOPRNS_1:27;
  a*z + b*z = (a+b)*z by RLVECT_1:def 6
    .= z by A1,RLVECT_1:def 8;
  hence thesis by A9;
end;
