 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 Th16:
  for s st s > 0 holds
     mlt(s,TOP-REAL n).: Ball(p,r)    = Ball(s*p,r*s) &
     mlt(s,TOP-REAL n).: cl_Ball(p,r) = cl_Ball(s*p,r*s) &
     mlt(s,TOP-REAL n).: Sphere(p,r)  = Sphere(s*p,r*s)
proof
   let s such that
       A1:s>0;
   set TR=TOP-REAL n,M=mlt(s,TR),ss=|.s.|,s1=1/s;
   A2:ss =s by A1,ABSVALUE:def 1;
   A4:dom M = [#]TR by FUNCT_2:def 1;
   A5:s*s1 = 1 by A1,XCMPLX_1:87;
   thus M.: Ball(p,r) = Ball(s*p,r*s)
   proof
     thus M.: Ball(p,r) c= Ball(s*p,r*s)
     proof
       let y be object;
       assume y in M.:Ball(p,r);
       then consider x be object such that
           A6: x in dom M
        and
           A7: x in Ball(p,r)
        and
           A8: M.x = y by FUNCT_1:def 6;
       reconsider q=x as Point of TR by A6;
       A9: y = s*q by A8, RLTOPSP1:def 13;
        (s*q) - (s*p) =s*(q-p) by RLVECT_1:34;
       then A10: |. (s*q) - (s*p).| = s* |.q-p.| by A2,EUCLID:11;
        s*|.q-p.| < r*s by A7,TOPREAL9:7,A1,XREAL_1:68;
       hence thesis by A10,A9;
      end;
     let y be object;
     A11: r*s*s1 = r*(s*s1);
     assume
         A12:y in Ball(s*p,r*s);
     then reconsider q=y as Point of TR;
     A13: |.(s1*q)-p.|*s*s1 = |.(s1*q)-p.|*(s*s1);
     A14:s*(s1*q) = (s*s1)*q by RLVECT_1:def 7
         .= 1*q by A1,XCMPLX_1:87
         .= q by RLVECT_1:def 8;
     then |.q-s*p.| = |.s*((s1*q)-p).| by RLVECT_1:34
         .= s*|.(s1*q)-p.| by A2,EUCLID:11;
     then |.(s1*q)-p.| < r
       by A11,A13,A5,A12,TOPREAL9:7,A1,XREAL_1:68;
     then A15: s1*q in Ball(p,r);
      M.(s1*q) = q by A14,RLTOPSP1:def 13;
     hence thesis by A4,A15,FUNCT_1:def 6;
    end;
   thus M.: cl_Ball(p,r) = cl_Ball(s*p,r*s)
   proof
     thus M.: cl_Ball(p,r) c= cl_Ball(s*p,r*s)
     proof
       let y be object;
       assume y in M.:cl_Ball(p,r);
       then consider x be object such that
           A16: x in dom M
        and
           A17: x in cl_Ball(p,r)
        and
           A18: M.x = y by FUNCT_1:def 6;
       reconsider q=x as Point of TR by A16;
       A19: y = s*q by A18, RLTOPSP1:def 13;
        (s*q) - (s*p) =s*(q-p) by RLVECT_1:34;
       then A20: |. (s*q) - (s*p).| = s* |.q-p.| by A2,EUCLID:11;
        s*|.q-p.| <= r*s by A17,TOPREAL9:8,A1,XREAL_1:64;
       hence thesis by A20,A19;
      end;
     let y be object;
     A21: r*s*s1 = r*(s*s1);
     assume
         A22:y in cl_Ball(s*p,r*s);
     then reconsider q=y as Point of TR;
     A23: |.(s1*q)-p.|*s*s1 = |.(s1*q)-p.|*(s*s1);
     A24:s*(s1*q) = (s*s1)*q by RLVECT_1:def 7
         .= 1*q by A1,XCMPLX_1:87
         .= q by RLVECT_1:def 8;
     then |.q-s*p.| = |.s*((s1*q)-p).| by RLVECT_1:34
         .= s*|.(s1*q)-p.| by A2,EUCLID:11;
     then
       |.(s1*q)-p.| <= r by A21,A23,A5,A22,TOPREAL9:8,A1,XREAL_1:64;
     then A25: s1*q in cl_Ball(p,r);
      M.(s1*q) = q by A24,RLTOPSP1:def 13;
     hence thesis by A4,A25,FUNCT_1:def 6;
    end;
   thus M.: Sphere(p,r) c= Sphere(s*p,r*s)
   proof
     let y be object;
     assume y in M.:Sphere(p,r);
     then consider x be object such that
         A26: x in dom M
      and
         A27: x in Sphere(p,r)
      and
         A28: M.x = y by FUNCT_1:def 6;
     reconsider q=x as Point of TR by A26;
      (s*q) - (s*p) =s*(q-p) by RLVECT_1:34;
     then A29: |. (s*q) - (s*p).| = s* |.q-p.| by A2,EUCLID:11;
      s*|.q-p.| = r*s by A27,TOPREAL9:9;
     then s*q in Sphere(s*p,r*s) by A29;
     hence thesis by A28, RLTOPSP1:def 13;
    end;
   let y be object;
   assume
       A30:y in Sphere(s*p,r*s);
   then reconsider q=y as Point of TR;
   A31: |.(s1*q)-p.|*s*s1 = |.(s1*q)-p.|*(s*s1);
   A32: r*s*s1 = r*(s*s1);
   A33:s*(s1*q) = (s*s1)*q by RLVECT_1:def 7
       .= 1*q by A1,XCMPLX_1:87
       .= q by RLVECT_1:def 8;
   then |.q-s*p.| = |.s*((s1*q)-p).| by RLVECT_1:34
       .= s*|.(s1*q)-p.| by A2,EUCLID:11;
   then |.(s1*q)-p.| = r by A32, A31,A5, A30,TOPREAL9:9;
   then A34: s1*q in Sphere(p,r);
    M.(s1*q) = q by A33,RLTOPSP1:def 13;
   hence thesis by A4,A34,FUNCT_1:def 6;
end;
