reserve A for non empty closed_interval Subset of REAL;

theorem
for a,b,c be Real, f be Function of REAL,REAL st
a <= b & b < c & f is_integrable_on ['a,c'] & f | ['a,c'] is bounded &
for x be Real st x in ['a,b'] holds  f.x = 0
holds
centroid(f,['a,c']) = centroid(f,['b,c'])
proof
 let a,b,c be Real;
 let f be Function of REAL,REAL;
 assume that
 A1: a <= b and
 A2: b < c and
 A4: f is_integrable_on ['a,c'] and
 A5: f | ['a,c'] is bounded and
 A6: for x be Real st x in ['a,b'] holds f.x = 0;
 B1: a <= c by A1,A2,XXREAL_0:2;
 reconsider F=f as PartFunc of REAL,REAL;
 B2A: dom f = REAL by FUNCT_2:def 1;
 B5: F | ['a,c'] is bounded by A5;
 b in [.a,c.] by A1,A2; then
 B3: b in ['a,c'] by INTEGRA5:def 3,A1,A2,XXREAL_0:2;
 integral (f,['a,b']) = 0 by Lm4,B2A,A6; then
 Q2A: integral (f,a,b) = 0 by INTEGRA5:def 4, A1;
 D2: integral (f,['a,c'])
  = integral (f,a,c) by INTEGRA5:def 4,A1,A2,XXREAL_0:2
 .= 0+(integral (f,b,c)) by INTEGRA6:17,A4,B5,B2A,B3,B1,Q2A
 .= integral (f,['b,c']) by INTEGRA5:def 4,A2;
 reconsider xf = (id REAL)(#)F as PartFunc of REAL,REAL;
 E2: dom xf = REAL by FUNCT_2:def 1;
 H1:REAL = dom (id REAL);
 reconsider iF = id REAL as PartFunc of REAL,REAL;
 H2: iF is_integrable_on ['a,c'] by Lm2;
 H3: iF | ['a,c'] is bounded by Lm2; then
 E4: iF (#) F is_integrable_on ['a,c'] by B2A,A4,B5,H1,H2,INTEGRA6:14;
 E5: ((id REAL)(#)F) | ['a,c'] is bounded by H3,B5,INTEGRA6:13;
 D1: xf is_integrable_on ['a,b'] & xf is_integrable_on ['b,c'] &
  integral (xf,a,c) = (integral (xf,a,b)) + (integral (xf,b,c))
     by INTEGRA6:17,E4,E5,E2,B3,B1;
 A7: dom ((id REAL)(#)f) = REAL by FUNCT_2:52;
 for x being Real st x in ['a,b'] holds xf.x = 0
 proof
  let x be Real;
  assume R1: x in ['a,b'];
  thus xf.x = ((id REAL).x) * (f.x) by VALUED_1:5
  .= ((id REAL).x) * 0 by R1,A6
  .= 0;
 end; then
 Q1: integral (xf,['a,b']) = 0 by A7,Lm4;
 D3:integral (xf,['a,c'])
  = integral (xf,a,c) by INTEGRA5:def 4,A1,A2,XXREAL_0:2
 .= 0 + (integral (xf,b,c)) by Q1,D1,INTEGRA5:def 4,A1
 .= (integral (xf,['b,c'])) by A2,INTEGRA5:def 4;
 centroid(f,['a,c'])
 = integral((id REAL)(#)f,['a,c'])/(integral (f,['b,c'])) by D2;
 hence thesis by D3;
end;
