reserve A for non empty closed_interval Subset of REAL;

theorem Th16X:
for r1,r2 be Real, f be Function of REAL,REAL st
f is_integrable_on A & f | A is bounded
holds
min(AffineMap(0,r1),(r2 (#) f)) is_integrable_on A &
min(AffineMap(0,r1),(r2 (#) f)) | A is bounded
proof
 let r1,r2 be Real, f be Function of REAL,REAL;
 assume that
 A2: f is_integrable_on A and
 A3: f | A is bounded;
 A c= REAL; then
 Df: A c= dom f by FUNCT_2:def 1;
 A c= REAL; then
 B21: A c= dom (AffineMap(0,r1)) by FUNCT_2:def 1;
 set F3 = AffineMap (0,r1);
 reconsider f,F3 as PartFunc of REAL,REAL;
 C1: (F3 | A) is continuous;
 A23: f is_integrable_on A & f | A is bounded by A2,A3;
 reconsider f,F3 as Function of REAL,REAL;
 F3 is_integrable_on A & F3 | A is bounded
 & (r2 (#) f) is_integrable_on A & (r2 (#) f) | A is bounded
  by INTEGRA6:9,Df,A23,RFUNCT_1:80,C1,INTEGRA5:10,INTEGRA5:11,B21;
 hence thesis by Th12;
end;
