reserve A for non empty closed_interval Subset of REAL;

theorem integra51011:
for A being non empty closed_interval Subset of REAL,
f be Function of REAL,REAL st
f is continuous holds f is_integrable_on A & f | A is bounded
proof
 let A being non empty closed_interval Subset of REAL,
 f be Function of REAL,REAL;
 assume A1: f is continuous;
DD: REAL = dom f by FUNCT_2:def 1;
 reconsider ff = f as PartFunc of REAL,REAL;
 ff | A is continuous by A1;
hence f is_integrable_on A & f | A is bounded by INTEGRA5:11,INTEGRA5:10,DD;
end;
