reserve a,b,c,d,e,x,r for Real,
  A for non empty closed_interval Subset of REAL,
  f,g for PartFunc of REAL,REAL;

theorem Th26:
  for a,b,r be Real, f be PartFunc of REAL,REAL
  st a <= b & ['a,b'] c= dom f
   & for t be Real st t in ['a,b'] holds f.t = r
  holds
    f | ['a,b'] is continuous
  & f | ['a,b'] is bounded
  & f is_integrable_on ['a,b']
  & integral(f,a,b) = r*(b-a)
proof
  let a,b,r be Real, f be PartFunc of REAL,REAL;
  assume
  A1: a <= b & ['a,b'] c= dom f
    & for t be Real st t in ['a,b'] holds f.t = r;
  A2: dom(f | ['a,b']) = ['a,b'] by A1,RELAT_1:62;

  for y be object
  holds y in rng(f | ['a,b']) iff y in {r}
  proof
    let y be object;
    hereby
      assume y in rng(f | ['a,b']);
      then consider x be object such that
      A3: x in dom(f | ['a,b'])
        & y = (f | ['a,b']).x by FUNCT_1:def 3;
      reconsider x as Real by A2,A3;
      y = f.x by A2,A3,FUNCT_1:49
       .= r by A1,A2,A3;
      hence y in {r} by TARSKI:def 1;
    end;
    assume y in {r}; then
    A4: y = r by TARSKI:def 1;
A5: ['a,b'] = [.a,b.] by A1,INTEGRA5:def 3; then
    y = f.a by A1,A4,XXREAL_1:1
     .= (f | ['a,b']).a by A1,A5,FUNCT_1:49,XXREAL_1:1;
    hence y in rng(f | ['a,b']) by A1,A2,A5,FUNCT_1:3,XXREAL_1:1;
  end; then
  A7: rng(f | ['a,b']) = {r} by TARSKI:2;
  rng(f | ['a,b']) c= REAL; then
  reconsider g = f || ['a,b'] as Function of ['a,b'],REAL by A2,FUNCT_2:2;
  integral(g) = r * vol([' a,b ']) by A7,INTEGRA4:4
             .= r * (b-a) by A1,Th5; then
  integral(f,['a,b']) = r * (b-a);
  hence thesis by A1,A7,FCONT_1:39,INTEGRA5:10,11,def 4;
end;
