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

theorem Th3:
  for Y be RealNormSpace,
      f be continuous PartFunc of REAL,the carrier of Y
    st a <= b & ['a,b'] c= dom f holds ||.f.|| | ['a,b'] is bounded
proof
   let Y be RealNormSpace, f be continuous PartFunc of REAL,the carrier of Y;
   assume A1: a <= b & [' a,b '] c= dom f;
P11: f | ['a,b'] is continuous;
   ['a,b'] c= dom ||.f.|| by NORMSP_0:def 3,A1;
   hence thesis by P11,A1,NFCONT_3:22,INTEGRA5:10;
end;
