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 Th1909:
  for f be continuous PartFunc of REAL,the carrier of Y
    st a <= c & c <= d & d <= b & ['a,b'] c= dom f
      holds f is_integrable_on ['c,d']
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; then
A2:c <= b & a <= d by XXREAL_0:2; then
A4:c in ['a,b'] & d in ['a,b'] by A1,INTEGR19:1;
   c = min(c,d) & d = max(c,d) by A1,XXREAL_0:def 9,def 10; then
   ['c,d'] c= ['a,b'] by A2,A1,XXREAL_0:2,A4,Lm2; then
   ['c,d'] c= dom f by A1;
   hence f is_integrable_on ['c,d'] by A1,INTEGR20:19;
end;
