reserve Z for RealNormSpace;
reserve a,b,c,d,e,r for Real;
reserve A,B for non empty closed_interval Subset of REAL;

theorem Th1915a:
  for f be PartFunc of REAL,the carrier of Z
    st f|A is bounded & B c= A & B c= dom (f|A)
      holds f|B is bounded
proof
  let f be PartFunc of REAL,the carrier of Z;
  assume
A1: f|A is bounded & B c= A & B c= dom (f|A);
  set g = f|A;
A4: g|B is bounded by Th1915,A1;
  consider r be Real such that
A5: for x be set st x in dom (g|B) holds ||. (g|B)/.x .|| <r by A4;
  g|B = f|B by A1,RELAT_1:74;
  hence f|B is bounded by A5;
end;
