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

theorem Th30:
  for g be continuous PartFunc of REAL,REAL n
     st dom g =[' a,b ']
   holds g is_integrable_on [' a,b ']
proof
  let g be continuous PartFunc of REAL,REAL n;
  assume A1: dom g =[' a,b '];
  let i be Element of NAT;
  assume i in Seg n; then
A2: proj(i,n)*g is continuous by NFCONT_4:44;
  dom (proj(i,n))=REAL n by FUNCT_2:def 1;
  then rng g c= dom(proj(i,n));
  then
A3: [' a,b '] c= dom (proj(i,n)*g) by A1,RELAT_1:27;
  (proj(i,n)*g) | [' a,b '] is continuous by A2;
  hence thesis by A3,INTEGRA5:11;
end;
