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 Th51:
  F is Cover of Closed-Interval-TSpace(r,s) & F is open connected
  & r <= s implies 1 <= len C
proof
  assume that
A1: F is Cover of Closed-Interval-TSpace(r,s) & F is open & F is connected and
A2: r <= s;
  assume not thesis;
  then len C+1 <= 0+1 by NAT_1:13;
  then
A3: C is empty by XREAL_1:6;
  union rng C = [.r,s.] by A1,A2,Def2;
  hence thesis by A2,A3,RELAT_1:38,XXREAL_1:1,ZFMISC_1:2;
end;
