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 Th1925:
  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 integral(r(#)f,c,d) = r*integral(f,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 A2: not c <= d;
    reconsider A = ['d,c'] as non empty closed_interval Subset of REAL;
    d = min(c,d) & c = max(c,d) by A2,XXREAL_0:def 9,def 10; then
    ['d,c'] c= ['a,b'] by A1,Lm2; then
A7: A c= dom f by A1; then
    A c= dom(r(#)f) by VFUNCT_1:def 4; then
A8: -integral(r(#)f,d,c) = integral(r(#)f,c,d)
  & -integral(f,d,c) = integral(f,c,d) by A2,A7,Th1947;
    integral(r(#)f,d,c) = r*integral(f,d,c) by A1,A2,Th1925a;
    hence integral(r(#)f,c,d) = r*integral(f,c,d) by A8,RLVECT_1:25;
   end;
   suppose c <= d;
    hence thesis by A1,Th1925a;
   end;
end;
