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 Th1915b:
  for f be PartFunc of REAL,the carrier of Z
    st a <= c & c <= d & d <= b & f| ['a,b'] is bounded & ['a,b'] c= dom f
      holds f| ['c,d'] is bounded
proof
   let f be PartFunc of REAL,the carrier of Z;
   assume A1:a <= c & c <= d & d <= b &
   f| ['a,b'] is bounded & ['a,b'] c= dom f;
A2:c <= b & a <= d by A1,XXREAL_0:2; then
A4:c in ['a,b'] & d in ['a,b'] by A1,INTEGR19:1;
   reconsider A = ['a,b'], B = ['c,d']
     as non empty closed_interval Subset of REAL;
   c = min(c,d) & d = max(c,d) by A1,XXREAL_0:def 9,def 10; then
A5:B c= A by A2,A1,XXREAL_0:2,A4,Lm2;
   then B c= dom(f|A) by A1,RELAT_1:62;
   hence f| ['c,d'] is bounded by A1,A5,Th1915a;
end;
