 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 Th15:
  transl(p,TOP-REAL n).: Ball(q,r)    = Ball(q+p,r) &
  transl(p,TOP-REAL n).: cl_Ball(q,r) = cl_Ball(q+p,r) &
  transl(p,TOP-REAL n).: Sphere(q,r)  = Sphere(q+p,r)
proof
  set TR=TOP-REAL n,T=transl(p,TR);
  A2:now let q;
      q-p+p = q -(p-p) by RLVECT_1:29
         .= q-0.TR by RLVECT_1:def 10
         .= q by RLVECT_1:13;
     hence transl(p,TR).(q-p)= q by RLTOPSP1:def 10;
   end;
   A3:now
     let x be Point of TR;
     thus (x+p) - (q+p) = ((x+p) - p)-q by RLVECT_1:27
         .=(x+(p - p))-q by RLVECT_1:28
         .= (x+0.TR)-q by RLVECT_1:def 10
         .= x- q by RLVECT_1:def 4;
    end;
   A4:dom T = [#]TR by FUNCT_2:def 1;
   thus T.: Ball(q,r) = Ball(q+p,r)
   proof
     thus T.: Ball(q,r) c= Ball(q+p,r)
     proof
       let y be object;
       assume y in transl(p,TR).:Ball(q,r);
       then consider x be object such that
           A5: x in dom T
        and
           A6: x in Ball(q,r)
        and
           A7: T.x = y by FUNCT_1:def 6;
       reconsider x as Point of TR by A5;
       A8: y = x+p by A7,RLTOPSP1:def 10;
       A9: (x+p) - (q+p) =x-q by A3;
        |.x-q.| < r by A6,TOPREAL9:7;
       hence thesis by A9,A8;
      end;
     let y be object;
     assume
         A10:y in Ball(q+p,r);
     then reconsider y as Point of TR;
     A11: (y-p) - q = y-(p + q ) by RLVECT_1:27;
      |. y-(q+p).| < r by A10,TOPREAL9:7;
     then A12: y-p in Ball(q,r) by A11;
      T.(y-p)= y by A2;
     hence thesis by A12,A4,FUNCT_1:def 6;
    end;
   thus T.: cl_Ball(q,r) = cl_Ball(q+p,r)
   proof
     thus T.: cl_Ball(q,r) c= cl_Ball(q+p,r)
     proof
       let y be object;
       assume y in T.:cl_Ball(q,r);
       then consider x be object such that
           A13: x in dom T
        and
           A14: x in cl_Ball(q,r)
        and
           A15: T.x = y by FUNCT_1:def 6;
       reconsider x as Point of TR by A13;
       A16: y = x+p by A15,RLTOPSP1:def 10;
       A17: (x+p) - (q+p) =x-q by A3;
        |.x-q.| <= r by A14,TOPREAL9:8;
       hence thesis by A17,A16;
      end;
     let y be object;
     assume
         A18:y in cl_Ball(q+p,r);
     then reconsider y as Point of TR;
     A19: (y-p) - q = y-(p + q ) by RLVECT_1:27;
      |. y-(q+p).| <= r by A18,TOPREAL9:8;
     then A20: y-p in cl_Ball(q,r) by A19;
      T.(y-p)= y by A2;
     hence thesis by A20,A4,FUNCT_1:def 6;
    end;
   thus T.: Sphere(q,r) c= Sphere(q+p,r)
   proof
     let y be object;
     assume y in T.:Sphere(q,r);
     then consider x be object such that
         A21: x in dom T
      and
         A22: x in Sphere(q,r)
      and
         A23: T.x = y by FUNCT_1:def 6;
     reconsider x as Point of TR by A21;
     A24: y = x+p by A23,RLTOPSP1:def 10;
     A25: (x+p) - (q+p) = x- q by A3;
      |.x-q.| = r by A22,TOPREAL9:9;
     hence thesis by A25,A24;
    end;
   let y be object;
   assume
       A26:y in Sphere(q+p,r);
   then reconsider y as Point of TR;
   A27: (y-p) - q = y-(p + q ) by RLVECT_1:27;
    |. y-(q+p).| = r by A26,TOPREAL9:9;
   then A28: y-p in Sphere(q,r) by A27;
    T.(y-p)= y by A2;
   hence thesis by A28,A4,FUNCT_1:def 6;
end;
