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
  for f be continuous PartFunc of REAL,the carrier of Y
    st a <= c & c <= d & d <= b & f is_integrable_on ['a,b']
  & f| ['a,b'] is bounded & ['a,b'] c= dom f
  holds -f is_integrable_on ['c,d'] & (-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 & f is_integrable_on ['a,b']
      & f| ['a,b'] is bounded & ['a,b'] c= dom f;
   -f = (-1)(#)f by VFUNCT_1:23;
   hence -f is_integrable_on ['c,d'] by A1,Th1911;
   f| ['c,d'] is bounded by A1,Th1915b;
   hence (-f) | ['c,d'] is bounded by Th1935b;
end;
