reserve c,c1,c2,x,y,z,z1,z2 for set;
reserve C1,C2,C3 for non empty set;

theorem Th1:
  for F be Membership_Func of C1 holds rng F is real-bounded & (for x st
x in dom F holds F.x <= upper_bound rng F) & for x st x in dom F
holds F.x >= lower_bound rng F
proof
  let F be Membership_Func of C1;
A1: [.0,jj.] is non empty closed_interval Subset of REAL by MEASURE5:14;
A2: rng F c= [.0,1.] by RELAT_1:def 19;
  then
A3: rng F is real-bounded by A1,XXREAL_2:45;
A4: for x st x in dom F holds F.x >= lower_bound rng F
  proof
    let x;
    assume x in dom F;
    then
A5: F.x in rng F by FUNCT_1:def 3;
    rng F is bounded_below by A3,XXREAL_2:def 11;
    hence thesis by A5,SEQ_4:def 2;
  end;
  for x st x in dom F holds F.x <= upper_bound rng F
  proof
    let x;
    assume x in dom F;
    then
A6: F.x in rng F by FUNCT_1:def 3;
    rng F is bounded_above by A3,XXREAL_2:def 11;
    hence thesis by A6,SEQ_4:def 1;
  end;
  hence thesis by A2,A1,A4,XXREAL_2:45;
end;
