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,B being non empty closed_interval Subset of
  REAL st A c= dom f & f|A is continuous & B c= A holds f is_integrable_on B
proof
  let f be PartFunc of REAL,REAL,
      A,B be non empty closed_interval Subset of REAL such
  that
A1: A c= dom f and
A2: f|A is continuous and
A3: B c= A;
  f|B is continuous by A2,A3,FCONT_1:16;
  hence thesis by A1,A3,Th11,XBOOLE_1:1;
end;
