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

theorem Th32:
  for f be continuous PartFunc of REAL,REAL-NS n st dom f = [' a,b ']
   holds f|([' a,b ']) is bounded
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;
  g is continuous by NFCONT_4:23;
  then g|([' a,b ']) is bounded by A1,Th29;
  hence f|([' a,b ']) is bounded by INTEGR19:34;
end;
