reserve X, Y for RealNormSpace;

theorem Th12:
  for x be Point of X,r be Real, V be Subset of
  LinearTopSpaceNorm X st V = Ball(x,r) holds V is convex
proof
  let x be Point of X,r be Real, V be Subset of LinearTopSpaceNorm X;
  reconsider V1 = Ball(x,r) as Subset of X;
A1: for u,v being Point of X, s being Real
  st 0 < s & s <1 & u in V1 & v in
  V1 holds s*u + (1-s)*v in V1
  proof
    let u,v be Point of X, s be Real;
    assume that
A2: 0 < s and
A3: s < 1 and
A4: u in V1 and
A5: v in V1;
A6: ex v1 be Point of X st v =v1 & ||.x-v1.||<r by A5;
    0 < |.s.| & ex u1 be Point of X st u =u1 & ||.x-u1.||<r by A2,A4,
ABSVALUE:def 1;
    then
A7: |.s.|*||.x-u.|| < |.s.|*r by XREAL_1:68;
    s+0<1 by A3;
    then
A8: 1-s >0 by XREAL_1:20;
    then 0 < |.1-s.| by ABSVALUE:def 1;
    then |.1-s.|*||.x-v.|| < |.1-s.|*r by A6,XREAL_1:68;
    then
    |.s.|*||.x-u.|| + |.1-s.|*||.x-v.|| < |.s.|*r + |.1-s.|*r by A7,
XREAL_1:8;
    then |.s.|*||.x-u.|| + |.1-s.|*||.x-v.|| < s*r + |.1-s.|*r by A2,
ABSVALUE:def 1;
    then |.s.|*||.x-u.|| + |.1-s.|*||.x-v.|| < s*r + (1-s)*r by A8,
ABSVALUE:def 1;
    then ||.s*(x-u).|| + |.1-s.|*||.x-v.|| < 1* r by NORMSP_1:def 1;
    then
A9: ||.s*(x-u).|| + ||.(1-s)*(x-v).|| < r by NORMSP_1:def 1;
    ||.s*x +(1-s)*x -(s*u + (1-s)*v).|| = ||.s*x +(-(s*u + (1-s)*v))+(1-s)
    *x.|| by RLVECT_1:def 3
      .= ||.s*x +(-1)*(s*u + (1-s)*v) +(1-s)*x.|| by RLVECT_1:16
      .= ||.s*x + ((-1)*(s*u) +(-1)*((1-s)*v)) +(1-s)*x.|| by RLVECT_1:def 5
      .= ||.s*x + (-s*u +(-1)*((1-s)*v)) +(1-s)*x.|| by RLVECT_1:16
      .= ||.s*x + (-s*u +-(1-s)*v) + (1-s)*x.|| by RLVECT_1:16
      .= ||.(s*x + -s*u) +-(1-s)*v + (1-s)*x.|| by RLVECT_1:def 3
      .= ||.s*x - s*u + ((1-s)*x -(1-s)*v).|| by RLVECT_1:def 3
      .= ||.s* (x - u) + ((1-s)*x -(1-s)*v).|| by RLVECT_1:34
      .= ||.s* (x - u) + (1-s)*(x -v).|| by RLVECT_1:34;
    then
    ||.s*x +(1-s)*x -(s*u + (1-s)*v).|| <= ||.s*(x-u).|| + ||.(1-s)*(x-v)
    .|| by NORMSP_1:def 1;
    then ||.s*x +(1-s)*x -(s*u + (1-s)*v).|| < r by A9,XXREAL_0:2;
    then ||.(s+(1-s))*x -(s*u + (1-s)*v).|| < r by RLVECT_1:def 6;
    then ||.x -(s*u + (1-s)*v).|| < r by RLVECT_1:def 8;
    hence thesis;
  end;
  assume
A10: V = Ball(x,r);
  for u,v being Point of LinearTopSpaceNorm X,
       s being Real st 0 < s & s
  < 1 & u in V & v in V holds s*u + (1-s)*v in V
  proof
    let u,v being Point of LinearTopSpaceNorm X;
    let s being Real;
    reconsider u1=u as Point of X by NORMSP_2:def 4;
    reconsider v1=v as Point of X by NORMSP_2:def 4;
    s*u1 = s*u & (1-s)*v1 =(1-s)*v by NORMSP_2:def 4;
    then
A11: s*u1 + (1-s)*v1 = s*u + (1-s)*v by NORMSP_2:def 4;
    assume 0 < s & s < 1 & u in V & v in V;
    hence thesis by A10,A1,A11;
  end;
  hence thesis;
end;
