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

theorem Th41:
  for jauge being Function of [.r,s.],].0,+infty.[,
  S being Subset-Family of Closed-Interval-TSpace(r,s)
  st S = the set of all ].x-jauge.x,x+jauge.x.[ /\ [.r,s.] where
  x is Element of [.r,s.] holds S is connected
  proof
    let jauge be Function of [.r,s.],].0,+infty.[,
    S be Subset-Family of Closed-Interval-TSpace(r,s);
    assume
A1: S = the set of all ].x-jauge.x,x+jauge.x.[ /\ [.r,s.] where
    x is Element of [.r,s.];
    for X being Subset of Closed-Interval-TSpace(r,s) st X in S
    holds X is connected
    proof
      let X be Subset of Closed-Interval-TSpace(r,s);
      assume X in S;
      then consider x0 be Element of [.r,s.] such that
A2:   X = ].x0-jauge.x0,x0+jauge.x0.[ /\ [.r,s.] by A1;
      thus thesis by A2,RCOMP_3:43;
    end;
    hence thesis by RCOMP_3:def 1;
  end;
