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 Th57:
  F is Cover of Closed-Interval-TSpace(r,s) & F is open connected
  & r <= s implies r in C/.1
proof
  assume that
A1: F is Cover of Closed-Interval-TSpace(r,s) & F is open & F is connected and
A2: r <= s;
  1 <= len C by A1,A2,Th51;
  then
A3: C.1 = C/.1 by FINSEQ_4:15;
  per cases;
  suppose
    [.r,s.] in F;
    then C = <*[.r,s.]*> by A1,A2,Def2;
    then C/.1 = [.r,s.] by FINSEQ_4:16;
    hence thesis by A2,XXREAL_1:1;
  end;
  suppose
    not [.r,s.] in F;
    then ex p being Real st r < p & p <= s & C.1 = [.r,p.[ by A1,A2,Def2;
    hence thesis by A3,XXREAL_1:3;
  end;
end;
