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 Th1911:
  for f be continuous PartFunc of REAL,the carrier of Y
    st a <= c & c <= d & d <= b & ['a,b'] c= dom f
  holds r(#)f is_integrable_on ['c,d'] & (r(#)f) | ['c,d'] is bounded
proof
   let f be continuous PartFunc of REAL,the carrier of Y;
   assume A1: a <= c & c <= d & d <= b  & ['a,b'] c= dom f;
   reconsider A = ['c,d'] as non empty closed_interval Subset of REAL;
A2:f is_integrable_on A by A1,Th1909;
   A c= dom f by A1,INTEGR19:2;
   hence r(#)f is_integrable_on ['c,d'] by A2,INTEGR18:13;
   a <= d by A1,XXREAL_0:2; then
   f| ['a,b'] is bounded by A1,Th1,XXREAL_0:2;
   then f| ['c,d'] is bounded by A1,Th1915b;
   hence (r(#)f) | ['c,d'] is bounded by Th1935a;
end;
