reserve n,m,i,k for Element of NAT;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,x0,x1,x2 for Real;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve h for PartFunc of REAL,REAL-NS n;
reserve W for non empty set;

theorem Th4:
 for r be Real, z be Element of REAL n,w be Point of REAL-NS n
  st z=w holds
 {y where y is Element of REAL n: |.y-z.| < r} =
 {y where y is Point of REAL-NS n: ||.y-w.|| < r}
proof
  let r be Real, z be Element of REAL n,w be Point of REAL-NS n;
  assume A1: z=w;
  set N1={y where y is Element of REAL n: |.y-z.| < r};
  set N2={y where y is Point of REAL-NS n: ||.y-w.|| < r};
A2: N1 c= N2
  proof let x be object;
    assume x in N1;
    then consider y be Element of REAL n such that
A3:   x=y & |.y-z.| < r;
    reconsider x1=x as Element of REAL n by A3;
    reconsider x2=x1 as Point of REAL-NS n by REAL_NS1:def 4;
    ||.x2-w.|| < r by A1,A3,REAL_NS1:1,5;
    hence x in N2;
  end;
  N2 c= N1
  proof let x be object;
    assume x in N2;
    then consider y be Point of REAL-NS n such that
A4:   x=y & ||.y-w.|| < r;
    reconsider x3=x as Point of REAL-NS n by A4;
    reconsider x4=x3 as Element of REAL n by REAL_NS1:def 4;
    |.x4-z.| < r by A1,A4,REAL_NS1:1,5;
    hence x in N1;
  end;
  hence N1 = N2 by A2,XBOOLE_0:def 10;
end;
