reserve A for non empty closed_interval Subset of REAL;

theorem
for f be Membership_Func of REAL holds f is bounded
proof
 let f be Membership_Func of REAL;
 ex r being Real st for x being set st x in dom f holds
  |. f.x .| < r
  proof
   take 1+1;
   thus for x being set st x in dom f holds
   |. f.x .| < 1+1
   proof
    let x be set;
    assume x in dom f; then
    reconsider x as Element of REAL;
    f.x = |.f.x.| by FUZZY_2:1,COMPLEX1:43; then
    |.f.x.|+0 < 1+1 by XREAL_1:8,FUZZY_2:1;
    hence thesis;
   end;
  end;
 hence thesis by COMSEQ_2:def 3;
end;
