reserve A for non empty closed_interval Subset of REAL;

theorem
for r be Real, f be Function of REAL,REAL st
r <> 0 & f is_integrable_on A & f | A is bounded
holds
centroid((r (#) f),A) = centroid(f,A)
proof
 let r be Real;
 let  f be Function of REAL,REAL;
 assume that
 A0: r <> 0 and
 A1: f is_integrable_on A and
 A2: f | A is bounded;
 A3: f | A is integrable by A1,INTEGRA5:def 1;
 A4: (f | A) | A is bounded by A2;
 B4: integral((r (#) f),A)
  = integral((r (#) f) | A) by INTEGRA5:def 2
 .= integral(r (#) (f|A)) by RFUNCT_1:49
 .= r * integral (f|A) by INTEGRA2:31,A3,A4
 .= r * integral(f,A) by INTEGRA5:def 2;
 B5: (((id REAL) (#) f) | A) is integrable &
  (((id REAL) (#) f) | A) | A is bounded
          by INTEGRA5:def 1,Lm3,A1,A2;
 integral((id REAL) (#) (r (#) f),A)
  = integral(r (#) ((id REAL) (#) f),A) by RFUNCT_1:13
 .= integral((r (#) ((id REAL) (#) f)) |A) by INTEGRA5:def 2
 .= integral((r (#) ((id REAL) (#) f) |A) ) by RFUNCT_1:49
 .= r * integral( ((id REAL) (#) f) |A ) by INTEGRA2:31,B5
 .= r * integral( (id REAL) (#) f,A ) by INTEGRA5:def 2;
 hence centroid((r (#) f),A)
  = centroid(f,A) by XCMPLX_1:91,A0,B4;
end;
