 reserve n for Nat;
 reserve s1 for sequence of Euclid n,
         s2 for sequence of REAL-NS n;
reserve r,s for Real;

theorem Th40:
  for jauge being Function of [.r,s.],].0,+infty.[,
  S being Subset-Family of Closed-Interval-TSpace(r,s) st r <= s &
  S = the set of all ].x-jauge.x,x+jauge.x.[ /\ [.r,s.] where
  x is Element of [.r,s.] holds S is open
  proof
    let jauge be Function of [.r,s.],].0,+infty.[,
    S be Subset-Family of Closed-Interval-TSpace(r,s);
    assume that
A1: r <= s and
A2: S = the set of all ].x-jauge.x,x+jauge.x.[ /\ [.r,s.] where
    x is Element of [.r,s.];
    for P be Subset of Closed-Interval-TSpace(r,s) st P in S holds P is open
    proof
      let P be Subset of Closed-Interval-TSpace(r,s);
      assume P in S;
      then consider x0 be Element of [.r,s.] such that
A4:   P = ].x0-jauge.x0,x0+jauge.x0.[ /\[.r,s.] by A2;
      set CIT = Closed-Interval-TSpace(r,s),
          CIM = Closed-Interval-MSpace(r,s);
A5:   CIT = TopSpaceMetr(CIM) by TOPMETR:def 7;
A6:   TopSpaceMetr(CIM) = TopStruct(# the carrier of CIM,Family_open_set CIM#)
        by PCOMPS_1:def 5;
      reconsider I = [.r,s.] as non empty Subset of RealSpace
        by A1,XXREAL_1:30;
      reconsider P1 = P as Subset of CIM by TOPMETR:def 7,A6;
      for t be Element of CIM st t in P1 holds ex r be Real st r > 0 &
      Ball(t,r) c= P1
      proof
        let t be Element of CIM;
        assume
A7:     t in P1;
        the carrier of CIM c= the carrier of RealSpace by TOPMETR:def 1;
        then reconsider tr = t as Point of RealSpace;
        reconsider XJ = ].x0-jauge.x0,x0+jauge.x0.[ as Subset of RealSpace;
        reconsider XK = XJ as Subset of R^1 by TOPMETR:17;
        [.r,s.] is non empty by A1,XXREAL_1:30;
        then x0 in [.r,s.];
        then reconsider p = x0 as Point of RealSpace;
        tr in XK by A7,A4,XBOOLE_0:def 4;
        then consider r0 be Real such that
A8:     r0 > 0 and
A9:     Ball(tr,r0) c= XK by JORDAN6:35,TOPMETR:15,TOPMETR:def 6;
        take r0;
        Ball(t,r0) = Ball(tr,r0) /\ the carrier of CIM by TOPMETR:9;
        then Ball(t,r0) = Ball(tr,r0) /\ [.r,s.] by A1,TOPMETR:10;
        hence thesis by A4,A8,A9,XBOOLE_1:27;
      end;
      then P in Family_open_set CIM by PCOMPS_1:def 4;
      hence thesis by A5,A6,PRE_TOPC:def 2;
    end;
    hence thesis by TOPS_2:def 1;
  end;
