reserve x,y,y1,y2 for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,h,g,h1 for Membership_Func of C;

theorem Th12:
  for a,b be Element of REAL, f be PartFunc of C,REAL st rng f c=
[.a,b.] & a <= b holds for x being Element of C st x in dom f holds a <= f.x &
  f.x <= b
proof
  let a,b be Element of REAL;
  let f be PartFunc of C,REAL;
  assume that
A1: rng f c= [.a,b.] and
A2: a <= b;
  for x being Element of C st x in dom f holds a <= f.x & f.x <= b
  proof
    reconsider A=[.a,b.] as
     non empty closed_interval Subset of REAL by A2,MEASURE5:14;
    let x be Element of C;
    A = [. lower_bound A, upper_bound A .] by INTEGRA1:4;
    then
A3: a=lower_bound A & b=upper_bound A by INTEGRA1:5;
    assume x in dom f;
    then f.x in rng f by FUNCT_1:def 3;
    hence thesis by A1,A3,INTEGRA2:1;
  end;
  hence thesis;
end;
