reserve a, b, r, s for Real;
reserve n, m for Nat,
  F for Subset-Family of Closed-Interval-TSpace (r,s);
reserve C for IntervalCover of F;

theorem Th52:
  F is Cover of Closed-Interval-TSpace(r,s) & F is open connected
  & r <= s & len C = 1 implies C = <*[.r,s.]*>
proof
  assume that
A1: F is Cover of Closed-Interval-TSpace(r,s) & F is open & F is
  connected & r <= s and
A2: len C = 1;
A3: union rng C = [.r,s.] by A1,Def2;
  C is non empty by A2;
  then rng C is non empty;
  then 1 in dom C by FINSEQ_3:32;
  then
A4: C.1 in rng C by FUNCT_1:def 3;
  C.1 = [.r,s.]
  proof
    thus for a being object st a in C.1 holds a in [.r,s.]
    by A3,A4,TARSKI:def 4;
    let a be object;
A5: dom C = {1} by A2,FINSEQ_1:2,def 3;
    assume a in [.r,s.];
    then consider Z being set such that
A6: a in Z and
A7: Z in rng C by A3,TARSKI:def 4;
    ex x being object st x in dom C & C.x = Z by A7,FUNCT_1:def 3;
    hence thesis by A6,A5,TARSKI:def 1;
  end;
  hence thesis by A2,FINSEQ_1:40;
end;
