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 Th1:
  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;
   set h = ||.f.|| | ['a,b'];
   set g = f|['a,b'];
   assume A1: a <= b & ['a,b'] c= dom f;
   then h is bounded by Th3; then
   consider r be Real such that
P2: for y being set st y in dom h holds |. h.y .| < r by COMSEQ_2:def 3;
D1:dom ||.f.|| = dom f by NORMSP_0:def 3;
   for y be set st y in dom g holds ||. g/.y .|| < r
   proof
    let y be set;
    assume P3: y in dom g; then
P5: y in ['a,b'] by A1,RELAT_1:62; then
P6: y in dom h by A1,D1,RELAT_1:62;
P81:h.y = ||.f.||.y by P5,FUNCT_1:49
       .= ||. f/.y .|| by NORMSP_0:def 2,D1,P5,A1;
    f/.y = f.y by PARTFUN1:def 6,P5,A1
        .= g.y by P5,FUNCT_1:49
        .= g/.y by PARTFUN1:def 6,P3;
    then |. h.y .| = ||. g/.y .|| by P81,ABSVALUE:def 1;
    hence thesis by P2,P6;
   end;
   hence thesis;
end;
