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

theorem Th29:
  for g be continuous PartFunc of REAL, REAL n st dom g = [' a,b ']
  holds g | [' a,b '] is bounded
proof
  let g be continuous PartFunc of REAL, REAL n;
  assume A1: dom g =[' a,b '];
A2: for i be Element of NAT st i in Seg n holds
  (proj(i,n)*g) | [' a,b '] is continuous
  proof
    let i be Element of NAT;
    assume i in Seg n; then
    proj(i,n)*g is continuous by NFCONT_4:44;
    hence (proj(i,n)*g) | [' a,b '] is continuous;
  end;
  let i be Element of NAT;
  assume A3: i in Seg n;
  dom (proj(i,n))=REAL n by FUNCT_2:def 1;
  then rng g c= dom(proj(i,n));
  then [' a,b '] c= dom (proj(i,n)*g) by A1,RELAT_1:27;
  then (proj(i,n)*g) | [' a,b ']  is bounded by A3,A2,INTEGRA5:10;
  hence thesis by Th28;
end;
