 reserve a,b,c,x for Real;
 reserve C for non empty set;

theorem HgtBnd:
  for F being FuzzySet of C holds
    0 <= height F & height F <= 1
  proof
    let F be FuzzySet of C;
B0: 0 is LowerBound of rng F
    proof
      let x be ExtReal;
      assume x in rng F; then
      consider xx being object such that
B1:   xx in dom F & x = F.xx by FUNCT_1:def 3;
      reconsider xx as Element of C by B1;
      thus thesis by B1,FUZZY_2:1;
    end;
b2: 1 is UpperBound of rng F
    proof
      let x be ExtReal;
      assume x in rng F; then
      consider xx being object such that
B1:   xx in dom F & x = F.xx by FUNCT_1:def 3;
      reconsider xx as Element of C by B1;
      thus thesis by B1,FUZZY_2:1;
    end;
    inf rng F <= sup rng F by XXREAL_2:40;
    hence thesis by b2,B0,XXREAL_2:def 4,XXREAL_2:def 3;
  end;
