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,g be continuous PartFunc of REAL,the carrier of Y
    st a <= c & c <= d & d <= b & ['a,b'] c= dom f & ['a,b'] c= dom g
  holds f-g is_integrable_on ['c,d'] & (f-g) | ['c,d'] is bounded
proof
   let f,g be continuous PartFunc of REAL,the carrier of Y;
   assume A1: a <= c & c <= d & d <= b & ['a,b'] c= dom f & ['a,b'] c= dom g;
   then a <= d by XXREAL_0:2;
   then
A1X:f is_integrable_on ['a,b'] & g is_integrable_on ['a,b']
  & f| ['a,b'] is bounded & g| ['a,b'] is bounded
       by A1,Th1,INTEGR20:19,XXREAL_0:2;
   reconsider A = ['c,d'] as non empty closed_interval Subset of REAL;
A2:f is_integrable_on A & g is_integrable_on A by A1,Th1909;
   A c= dom f & A c= dom g by A1,INTEGR19:2;
   hence f - g is_integrable_on ['c,d'] by A2,INTEGR18:15;
A3:f| ['c,d'] is bounded & g| ['c,d'] is bounded by A1,A1X,Th1915b;
   ['c,d'] /\ ['c,d'] = ['c,d'];
   hence (f - g) | ['c,d'] is bounded by A3,Th1935;
end;
