reserve i, j, n for Element of NAT,
  f, g, h, k for FinSequence of REAL,
  M, N for non empty MetrSpace;

theorem
  the carrier of M = the carrier of N & (for m being Point of M,
   n being Point of N, r being Real st r > 0 & m = n
  ex r1 being Real st r1 > 0 & Ball(n,
  r1) c= Ball(m,r)) & (for m being Point of M, n being Point of N,
    r being Real st r > 0 & m = n
 ex r1 being Real st r1 > 0 & Ball(m,r1) c= Ball(n,r)) implies
  TopSpaceMetr M = TopSpaceMetr N
proof
  assume that
A1: the carrier of M = the carrier of N and
A2: for m being Point of M, n being Point of N,
     r being Real st r > 0 &
  m = n ex r1 being Real st r1 > 0 & Ball(n,r1) c= Ball(m,r) and
A3: for m being Point of M, n being Point of N,
    r being Real st r > 0 &
  m = n ex r1 being Real st r1 > 0 & Ball(m,r1) c= Ball(n,r);
A4: Family_open_set M = Family_open_set N
  proof
    thus Family_open_set M c= Family_open_set N
    proof
      let X be object;
       reconsider XX=X as set by TARSKI:1;
      assume
A5:   X in Family_open_set M;
      for n being Point of N st n in XX
       ex r being Real st r > 0 & Ball(n,r) c= XX
      proof
        let n be Point of N such that
A6:     n in XX;
        reconsider m = n as Point of M by A1;
        consider r being Real such that
A7:     r > 0 and
A8:     Ball(m,r) c= XX by A5,A6,PCOMPS_1:def 4;
       consider r1 being Real such that
A9:     r1 > 0 & Ball(n,r1) c= Ball(m,r) by A2,A7;
        take r1;
        thus thesis by A8,A9;
      end;
      hence thesis by A1,A5,PCOMPS_1:def 4;
    end;
    let X be object;
     reconsider XX=X as set by TARSKI:1;
    assume
A10: X in Family_open_set N;
    for m being Point of M st m in XX
      ex r being Real st r > 0 & Ball(m,r) c= XX
    proof
      let m be Point of M such that
A11:  m in XX;
      reconsider n = m as Point of N by A1;
      consider r being Real such that
A12:  r > 0 and
A13:  Ball(n,r) c= XX by A10,A11,PCOMPS_1:def 4;
      consider r1 being Real such that
A14:  r1 > 0 & Ball(m,r1) c= Ball(n,r) by A3,A12;
      take r1;
      thus thesis by A13,A14;
    end;
    hence thesis by A1,A10,PCOMPS_1:def 4;
  end;
  TopSpaceMetr M = TopStruct (#the carrier of M, Family_open_set M#) by
PCOMPS_1:def 5;
  hence thesis by A1,A4,PCOMPS_1:def 5;
end;
