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

theorem Th39:
  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 Cover of Closed-Interval-TSpace(r,s)
  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.];
    [.r,s.] c= union S
    proof
      let x be object;
      assume
A3:   x in [.r,s.];
      then reconsider x0 = x as Element of [.r,s.];
      x0 in dom jauge by A3,PARTFUN1:def 2;
      then jauge.x0 in rng jauge by FUNCT_1:3;
      then 0 < jauge.x0 by XXREAL_1:4;
      then x0 - jauge.x0 < x0 - 0 & x0 + 0 < x0 + jauge.x0
        by XREAL_1:15,XREAL_1:8;
      then x0 in ].x0-jauge.x0,x0+jauge.x0.[ by XXREAL_1:4; then
A5:   x0 in ].x0-jauge.x0,x0+jauge.x0.[ /\ [.r,s.] by A3,XBOOLE_0:def 4;
      ].x0-jauge.x0,x0+jauge.x0.[ /\ [.r,s.] in S by A2;
      hence thesis by A5,TARSKI:def 4;
    end;
    then S is Cover of [.r,s.] by SETFAM_1:def 11;
    hence thesis by A1,TOPMETR:18;
  end;
