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 Th1922:
  for f be continuous PartFunc of REAL,the carrier of Y
    st a <= b & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b']
  holds ||.f.|| is_integrable_on ['min(c,d),max(c,d)']
   & ||.f.|| | ['min(c,d),max(c,d)'] is bounded
   & ||. integral(f,c,d) .|| <= integral(||.f.||,min(c,d),max(c,d))
proof
   let f be continuous PartFunc of REAL,the carrier of Y;
   assume A1: a <= b & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'];
   per cases;
   suppose A3: not c <= d; then
A5: d = min(c,d) & c = max(c,d) by XXREAL_0:def 9,def 10; then
    ['d,c'] c= dom f by A1,INTEGR19:3; then
    integral(f,c,d) = -integral(f,d,c) by A3,Th1947; then
    ||. integral(f,c,d) .|| = ||.integral(f,d,c).|| by NORMSP_1:2;
    hence thesis by A1,A3,A5,Lm10;
   end;
   suppose A6: c <= d; then
    c = min(c,d) & d = max(c,d) by XXREAL_0:def 9,def 10;
    hence thesis by A1,A6,Lm10;
   end;
end;
