
theorem Th13:
  for P being Subset of R^1 holds P is compact implies [#](P) is compact
proof
  let P be Subset of R^1;
  assume
A1: P is compact;
  now
    per cases;
    case
      [#](P) <> {};
      [#](P) is real-bounded by A1,Th11;
      hence thesis by A1,Th12,RCOMP_1:11;
    end;
    case
A2:   [#](P) = {};
      assume not [#](P) is compact;
      then
A3:   ex s1 being Real_Sequence st (rng s1) c= [#](P) & not(ex s2 being
Real_Sequence st s2 is subsequence of s1 & s2 is convergent & (lim s2) in [#](P
      )) by RCOMP_1:def 3;
      0 in NAT;
      hence thesis by A2,A3,FUNCT_2:4;
    end;
  end;
  hence thesis;
end;
