 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 Th17:
  for f be rotation homogeneous additive Function of
        TOP-REAL n,TOP-REAL n st f is onto
    holds  f.: Ball(p,r)    = Ball(f.p,r) &
           f.: cl_Ball(p,r) = cl_Ball(f.p,r) &
           f.: Sphere(p,r)  = Sphere(f.p,r)
proof
   set TR=TOP-REAL n;
   let f be rotation homogeneous additive Function of TR,TR;
   assume
       f is onto;
   A2: -f.p = (-1)*(f.p) by RLVECT_1:16
       .= f.((-1)*p) by TOPREAL9:def 4
       .= f.(-p) by RLVECT_1:16;
   A3:rng f = the carrier of TR by FUNCT_2:def 3;
   thus f.:Ball(p,r) = Ball(f.p,r)
   proof
     thus f.:Ball(p,r) c= Ball(f.p,r)
     proof
       let y be object;
       assume y in f.:Ball(p,r);
       then consider x be object such that
           A4: x in dom f
        and
           A5: x in Ball(p,r)
        and
           A6: f.x = y by FUNCT_1:def 6;
       reconsider q=x as Point of TR by A4;
        f.q-f.p = f.(q-p) by A2,VECTSP_1:def 20;
       then A7: |.f.q-f.p.|= |.q-p.| by MATRTOP3:def 4;
        |.q-p.| < r by A5,TOPREAL9:7;
       hence thesis by A7,A6;
      end;
     let y be object;
     assume
         A8:y in Ball(f.p,r);
     then consider x be object such that
         A9: x in dom f
      and
         A10: f.x=y by A3,FUNCT_1:def 3;
     reconsider x as Point of TR by A9;
      f.x-f.p = f.(x-p) by A2,VECTSP_1:def 20;
     then A11: |.f.x-f.p.|= |.x-p.| by MATRTOP3:def 4;
      |.f.x-f.p.| < r by A8,A10,TOPREAL9:7;
     then x in Ball(p,r) by A11;
     hence thesis by A9,A10,FUNCT_1:def 6;
    end;
   thus f.:cl_Ball(p,r) = cl_Ball(f.p,r)
   proof
     thus f.:cl_Ball(p,r) c= cl_Ball(f.p,r)
     proof
       let y be object;
       assume y in f.:cl_Ball(p,r);
       then consider x be object such that
           A12: x in dom f
        and
           A13: x in cl_Ball(p,r)
        and
           A14: f.x = y by FUNCT_1:def 6;
       reconsider q=x as Point of TR by A12;
        f.q-f.p = f.(q-p) by A2,VECTSP_1:def 20;
       then A15: |.f.q-f.p.|= |.q-p.| by MATRTOP3:def 4;
        |.q-p.| <= r by A13,TOPREAL9:8;
       hence thesis by A15,A14;
      end;
     let y be object;
     assume
         A16:y in cl_Ball(f.p,r);
     then consider x be object such that
         A17: x in dom f
      and
         A18: f.x=y by A3,FUNCT_1:def 3;
     reconsider x as Point of TR by A17;
      f.x-f.p = f.(x-p) by A2,VECTSP_1:def 20;
     then A19: |.f.x-f.p.|= |.x-p.| by MATRTOP3:def 4;
      |.f.x-f.p.| <= r by A16,A18,TOPREAL9:8;
     then x in cl_Ball(p,r) by A19;
     hence thesis by A17,A18,FUNCT_1:def 6;
    end;
   thus f.:Sphere(p,r) c= Sphere(f.p,r)
   proof
     let y be object;
     assume y in f.:Sphere(p,r);
     then consider x be object such that
         A20: x in dom f
      and
         A21: x in Sphere(p,r)
      and
         A22: f.x = y by FUNCT_1:def 6;
     reconsider q=x as Point of TR by A20;
      f.q-f.p = f.(q-p) by A2,VECTSP_1:def 20;
     then A23: |.f.q-f.p.|= |.q-p.| by MATRTOP3:def 4;
      |.q-p.| = r by A21,TOPREAL9:9;
     hence thesis by A23,A22;
    end;
   let y be object;
   assume
       A24:y in Sphere(f.p,r);
   then consider x be object such that
       A25: x in dom f
    and
       A26: f.x=y by A3,FUNCT_1:def 3;
   reconsider x as Point of TR by A25;
    f.x-f.p = f.(x-p) by A2,VECTSP_1:def 20;
   then A27: |.f.x-f.p.|= |.x-p.| by MATRTOP3:def 4;
    |.f.x-f.p.| = r by A24,A26,TOPREAL9:9;
   then x in Sphere(p,r) by A27;
   hence thesis by A25,A26,FUNCT_1:def 6;
end;
