reserve X, Y for RealNormSpace;

theorem Th13:
  for x be Point of X,r be Real, T be LinearOperator of X,Y
  , V be Subset of LinearTopSpaceNorm Y st V = T.: Ball(x,r) holds V is convex
proof
  let x be Point of X,r be Real, T be LinearOperator of X,Y, V be
  Subset of LinearTopSpaceNorm Y;
  reconsider V1 = T.: Ball(x,r) as Subset of Y;
A1: for u,v being Point of Y, s being Real
 st 0 < s & s <1 & u in V1 & v in
  V1 holds s*u + (1-s)*v in V1
  proof
    reconsider Bxr =Ball(x,r) as Subset of LinearTopSpaceNorm X by
NORMSP_2:def 4;
    let u,v being Point of Y, s being Real;
    assume that
A2: 0 < s & s < 1 and
A3: u in V1 and
A4: v in V1;
    consider z1 be object such that
A5: z1 in the carrier of X and
A6: z1 in Ball(x,r) and
A7: u= T.z1 by A3,FUNCT_2:64;
    reconsider zz1=z1 as Point of X by A5;
    reconsider za1=zz1 as Point of LinearTopSpaceNorm X by NORMSP_2:def 4;
    consider z2 be object such that
A8: z2 in the carrier of X and
A9: z2 in Ball(x,r) and
A10: v= T.z2 by A4,FUNCT_2:64;
    reconsider zz2=z2 as Point of X by A8;
A11: (1-s)*v=T.((1-s)*zz2) by A10,LOPBAN_1:def 5;
    reconsider za2=zz2 as Point of LinearTopSpaceNorm X by NORMSP_2:def 4;
    s*za1 =s*zz1 & (1-s)*za2 =(1-s)*zz2 by NORMSP_2:def 4;
    then
A12: s*za1 + (1-s)*za2=s*zz1 + (1-s)*zz2 by NORMSP_2:def 4;
    s*u=T.(s*zz1) by A7,LOPBAN_1:def 5;
    then
A13: s*u + (1-s)*v =T.(s*zz1 + (1-s)*zz2) by A11,VECTSP_1:def 20;
    Bxr is convex by Th12;
    then s*za1 + (1-s)*za2 in Bxr by A2,A6,A9;
    hence thesis by A12,A13,FUNCT_2:35;
  end;
  assume
A14: V = T.: Ball(x,r);
  for u,v being Point of LinearTopSpaceNorm Y,
      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 Y;
    let s be Real;
    reconsider u1=u as Point of Y by NORMSP_2:def 4;
    reconsider v1=v as Point of Y by NORMSP_2:def 4;
    s*u1 =s*u & (1-s)*v1 =(1-s)*v by NORMSP_2:def 4;
    then
A15: 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 A14,A1,A15;
  end;
  hence thesis;
end;
