reserve i,k,n,m for Element of NAT;
reserve a,b,r,r1,r2,s,x,x1,x2 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve X for set;

theorem
  for f being PartFunc of REAL,REAL,
  A being non empty closed_interval Subset of
  REAL, a,b st A=[.a,b.] holds integral(f,A)=integral(f,a,b)
proof
  let f be PartFunc of REAL,REAL;
  let A be non empty closed_interval Subset of REAL;
  let a,b;
  consider a1,b1 being Real such that
A1: a1 <= b1 and
A2: A = [.a1,b1.] by MEASURE5:14;
  assume A = [.a,b.];
  then
A3: a1=a & b1=b by A2,INTEGRA1:5;
  then integral(f,a,b)=integral(f,[' a,b ']) by A1,Def4;
  hence thesis by A1,A2,A3,Def3;
end;
