reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem
  for r being Real holds 0 <= r & r <= 1 iff r in the carrier of I[01]
proof
  let r be Real;
A1: [.0,1.] = { r1 where r1 is Real: 0 <= r1 & r1 <= 1 }
         by RCOMP_1:def 1;
  thus 0 <= r & r <= 1 implies r in the carrier of I[01] by A1,Th40;
  assume r in the carrier of I[01];
  then ex r2 being Real st r2 = r & 0 <= r2 & r2 <= 1 by A1,Th40;
  hence thesis;
end;
