reserve A for non empty closed_interval Subset of REAL;

theorem Lm3:
for f be Function of REAL,REAL st
f is_integrable_on A & f | A is bounded
holds
(id REAL) (#) f is_integrable_on A &
((id REAL) (#) f) | A is bounded
proof
 let f be Function of REAL,REAL;
 assume A1: f is_integrable_on A & f | A is bounded;
 A4: dom f = REAL by FUNCT_2:def 1;
 reconsider f as PartFunc of REAL,REAL;
 A2: f is_integrable_on A & f | A is bounded by A1;
 reconsider iR = (id REAL) as PartFunc of REAL,REAL;
 A3: A c= dom iR;
 B1: iR is_integrable_on A by Lm2;
 iR | A is bounded by Lm2;
 hence thesis by INTEGRA6:14,A2,A4,B1,A3,INTEGRA6:13;
end;
