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 Th1929:
  for f be PartFunc of REAL,the carrier of Y, E be Point of Y
    st ( a <= b & ['a,b'] c= dom f
     & for x be Real st x in ['a,b'] holds f/.x = E ) holds
   f is_integrable_on ['a,b'] & integral(f,a,b) = (b-a)*E
proof
   let f be PartFunc of REAL,the carrier of Y, E be Point of Y;
   assume that
A1: a <= b and
A2: ['a,b'] c= dom f and
A3: for x be Real st x in ['a,b'] holds f/.x = E;
   reconsider A=['a,b'] as non empty closed_interval Subset of REAL;
   f is Function of dom f, rng f by FUNCT_2:1; then
   f is Function of dom f,the carrier of Y by FUNCT_2:2; then
   reconsider g=f|A as Function of A,the carrier of Y by A2,FUNCT_2:32;
A7:{E} c= rng g
   proof
    let x be object;
    assume x in {E}; then
A5: x = E by TARSKI:def 1;
    consider a being Element of A such that
A6:  a in dom g by SUBSET_1:4;
    f/.a = E by A3; then
    f.a = E by A2,PARTFUN1:def 6; then
    g.a = E by FUNCT_1:49;
    hence thesis by A5,A6,FUNCT_1:3;
   end;
   rng g c= {E}
   proof
    let x be object;
    assume x in rng g;
    then consider a being Element of A such that
     a in dom g and
A9:  g.a=x by PARTFUN1:3;
    f.a=x by A9,FUNCT_1:49; then
    f/.a=x by A2,PARTFUN1:def 6; then
    x = E by A3;
    hence x in {E} by TARSKI:def 1;
   end; then
A10:rng g = {E} by A7,XBOOLE_0:def 10;
   hence f is_integrable_on ['a,b'] by Th404;
   integral g = (vol A) * E by A10,Th404; then
   integral(f,A) = (vol A) * E by A2,INTEGR18:def 8; then
   integral(f,A) = (b-a)*E by A1,INTEGRA6:5;
   hence integral(f,a,b) = (b-a)*E by A1,INTEGR18:def 9;
end;
