reserve A for non empty closed_interval Subset of REAL;

theorem
for f,g,h be Function of REAL,REAL, a,b,c be Real st
a <= b & b <= c & f is continuous & g is continuous &
h | [.a,c.] = f | [.a,b.] +* g | [.b,c.] &
integral(f,['a,b']) <> 0 & integral(g,['b,c']) <> 0 & f.b = g.b holds
centroid(h, ['a,c']) = 1 / integral(h,['a,c'])
 * ( centroid(f, ['a,b']) * integral(f,['a,b'])
   + centroid(g, ['b,c']) * integral(g,['b,c']) )
proof
 let f,g,h be Function of REAL,REAL, a,b,c be Real;
 assume that A1: a <= b & b <= c and
 A2: f is continuous & g is continuous and
 A3: h | [.a,c.] = f | [.a,b.] +* g | [.b,c.];
 assume A4:integral(f,['a,b']) <> 0 & integral(g,['b,c']) <> 0;
 assume
 A6:  f.b = g.b;
 A5: centroid(f, ['a,b']) * integral(f,['a,b'])
 + centroid(g, ['b,c']) * integral(g,['b,c'])
  = (integral((id REAL)(#)f,['a,b'])/integral(f,['a,b']))
   * integral(f,['a,b'])
   + centroid(g, ['b,c']) * integral(g,['b,c'] ) by FUZZY_6:def 1
 .= (integral((id REAL)(#)f,['a,b'])/integral(f,['a,b']))
   * integral(f,['a,b'])
   + (integral((id REAL)(#)g,['b,c'])/integral(g,['b,c']))
   * integral(g,['b,c'] ) by FUZZY_6:def 1
 .= integral((id REAL)(#)f,['a,b'])/(integral(f,['a,b'])
   / integral(f,['a,b']) )
   + (integral((id REAL)(#)g,['b,c'])/integral(g,['b,c']))
   * integral(g,['b,c'] ) by XCMPLX_1:82
 .= integral((id REAL)(#)f,['a,b'])/(1)
   + (integral((id REAL)(#)g,['b,c'])/integral(g,['b,c']))
   * integral(g,['b,c'] ) by XCMPLX_1:60,A4
 .= integral((id REAL)(#)f,['a,b'])/(1)
   + integral((id REAL)(#)g,['b,c'])/(integral(g,['b,c'])
   / integral(g,['b,c'])) by XCMPLX_1:82
 .= integral((id REAL)(#)f,['a,b'])
   + integral((id REAL)(#)g,['b,c'])/(1) by XCMPLX_1:60,A4
  .= integral((id REAL) (#) h,['a,c']) by FUZZY_7:43,A1,A2,A3,A6;
 centroid(h, ['a,c'])
 = integral((id REAL)(#)h,['a,c'])/integral(h,['a,c']) by FUZZY_6:def 1
 .= integral((id REAL)(#)h,['a,c'])*(1/integral(h,['a,c'])) by XCMPLX_1:99;
 hence thesis by A5;
end;
