reserve a,b,r for Real;
reserve A for non empty set;
reserve X,x for set;
reserve f,g,F,G for PartFunc of REAL,REAL;
reserve n for Element of NAT;

theorem Th19:
  a <= b & [. a,b .] c= X & X c= dom f & f|X is continuous implies
  f|[' a,b '] is continuous & f is_integrable_on [' a,b '] &
  f|[' a,b '] is bounded
proof
  assume a <= b; then
A1: [. a,b .] = [' a,b '] by INTEGRA5:def 3;
  assume that
A2: [. a,b .] c= X and
A3: X c= dom f and
A4: f|X is continuous;
  thus
A5: f|[' a,b '] is continuous by A1,A2,A4,FCONT_1:16;
  [. a,b .] c= dom f by A2,A3;
  hence thesis by A1,A5,INTEGRA5:10,17;
end;
