reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;

theorem Th33:
  for f be continuous PartFunc of REAL,REAL-NS n st dom f = [' a,b ']
  holds f is_integrable_on [' a,b ']
proof
  let f be continuous PartFunc of REAL,REAL-NS n;
  assume A1: dom f =[' a,b '];
  reconsider g = f as PartFunc of REAL, REAL n by REAL_NS1:def 4;
A2: g is continuous by NFCONT_4:23; then
A3: g is_integrable_on [' a,b '] by A1,Th30;
    g|([' a,b ']) is bounded by A2,A1,Th29;
  hence thesis by A1,A3,INTEGR19:44;
end;
