reserve A for non empty closed_interval Subset of REAL;

theorem
for r1,r2 be Real, f,F be Function of REAL,REAL st
f is_integrable_on A & f | A is bounded &
for x being Real holds F.x = min(r1, r2*(f.x))
holds
F is_integrable_on A & F | A is bounded
proof
 let r1,r2 be Real;
 let f,F be Function of REAL,REAL;
 assume that
 A2:  f is_integrable_on A & f | A is bounded and
 A4: for x being Real holds F.x = min(r1, r2*(f.x));
 DD: REAL = dom F
 & REAL = dom min(AffineMap(0,r1),(r2 (#) f)) by FUNCT_2:52;
 for x being object st x in dom F holds
 F . x = min(AffineMap(0,r1),(r2 (#) f)) . x
 proof
  let x be object;
  assume A0: x in dom F;
  reconsider x as Element of REAL by A0;
  min(AffineMap(0,r1),(r2 (#) f)) . x
   = min(AffineMap(0,r1).x,(r2 (#) f).x) by COUSIN2:def 1
  .= min(0*x+r1,(r2 (#) f).x) by FCONT_1:def 4
  .= min(r1,r2 * (f.x)) by VALUED_1:6;
  hence thesis by A4;
 end; then
 F = min(AffineMap(0,r1),(r2 (#) f)) by FUNCT_1:2,DD;
 hence thesis by A2,Th16X;
end;
