 reserve A for non empty Subset of REAL;

theorem Th15:
  for a,b,c be Real, f be Function of REAL,REAL st
  b > 0 & c > 0 &
  f | ['a-c,a+c'] =
    ( AffineMap ( b/c,b-a*b/c) |
  [. lower_bound ['a-c,a+c'], ((b+a*b/c) - (b-a*b/c))/((b/c)-(-b/c)) .] )
    +* ( AffineMap (-b/c,b+a*b/c) |
  [. (b+a*b/c - (b-a*b/c))/((b/c)-(-b/c)), upper_bound ['a-c,a+c'] .] )
    holds
  centroid (f,['a-c,a+c']) = a
proof
 let a,b,c be Real, f be Function of REAL,REAL;
 assume that
 A1: b > 0 and
 A2: c > 0 and
 A3: f | ['a-c,a+c'] = ( AffineMap ( b/c,b-a*b/c) |
   [. lower_bound ['a-c,a+c'], ((b+a*b/c) - (b-a*b/c))/((b/c)-(-b/c)) .] )
  +* ( AffineMap (-b/c,b+a*b/c) |
   [. ((b+a*b/c) - (b-a*b/c))/((b/c)-(-b/c)), upper_bound ['a-c,a+c'] .] );
 set s = b/c;
 set d = b-a*b/c;
 set p = -b/c;
 set q = b+a*b/c;
 set C = ['a-c,a+c'];
 A7:(q-d)/(s-p) = (b+a*b/c - b+a*b/c)/((b/c)+b/c)
 .= (a*b/c +a*(b/c))/(b/c+b/c) by XCMPLX_1:74
 .= (a*(b/c) +a*(b/c))/(b/c+b/c) by XCMPLX_1:74
 .= (a*(b/c + b/c))/(b/c+b/c)
 .= a*((b/c + b/c)/(b/c+b/c)) by XCMPLX_1:74
 .= a by XCMPLX_1:88,A1,A2;
 A10: a-c <= a & a <= a+c by XREAL_1:43,XREAL_1:29,A2;
 [.(lower_bound C),(upper_bound C).] = ['a-c,a+c'] &
 ['a-c,a+c'] = [.a-c,a+c.] by A10,INTEGRA5:def 3,INTEGRA1:4,XXREAL_0:2;
 then
 A8: lower_bound C = a-c & a+c = upper_bound C by INTEGRA1:5;
 Dd: d = (b*c-a*b)/c by XCMPLX_1:127,A2
   .= (c-a)*b/c
   .= (b/c)*(c-a) by XCMPLX_1:74;
 Qq: q = (a*b + b*c)/c by XCMPLX_1:113,A2
   .= b*(a + c)/c
   .=(b/c)*(a+c) by XCMPLX_1:74;
 CE: centroid(f,C) = (1/3*s*((a)^3 - (lower_bound C)^3)
  + 1/2*d*((a)^2 - (lower_bound C)^2)
  + 1/3*p*((upper_bound C)^3 - a^3)
  + 1/2*q*((upper_bound C)^2 - a^2) ) /
 ( 1/2*s*((a)^2 -(lower_bound C)^2) + d*(a - lower_bound C)
 + 1/2*p*((upper_bound C)^2 -(a)^2) + q*(upper_bound C -a))
 by FUZZY_6:48,A3,A7,Lm6,A2,A1
 .= ( 1/3*s*(a^3 - (a-c)^3) + 1/2*d*((a-(a-c))*(a+(a-c)))
      + 1/3*p*((a+c)^3 - a^3) + 1/2*q*(((a+c)-a)*((a+c)+a)) ) /
 ( 1/2*s*((a-(a-c))*(a+(a-c))) + d*c
 + 1/2*p*(((a+c)-a)*((a+c)+a)) + q*c ) by A8;
 B1: 1/3*s*(a^3 - (a-c)^3) + 1/2*d*(c*(a+a-c))
      + 1/3*p*((a+c)^3 - a^3) + 1/2*q*(c*(a+c+a))
 = (1/3*s*(a^3 - (a-c) ^3) + 1/2*d*(c*(a+a-c))
      + 1/3*p*((a+c) |^ 3 - a^3) + 1/2*q*(c*(a+c+a)) ) by POLYEQ_3:27
 .= (1/3*s*(a^3 - (a-c) |^3 ) + 1/2*d*(c*(a+a-c))
      + 1/3*p*((a+c) |^ 3 - a^3) + 1/2*q*(c*(a+c+a)) ) by POLYEQ_3:27
 .= (1/3*s*(a^3 - (a-c)^2*(a-c) ) + 1/2*c*d*(a+a-c)
      + 1/3*p*((a+c) |^ 3 - a^3) + 1/2*c*q*(a+c+a) ) by POLYEQ_3:27
 .= 1/3*(b/c)*(a^3 - (a-c)^2*(a-c) ) + 1/2*c*((b/c)*(c-a))*(a+a-c)
      + 1/3*(b/c)*(-1)*((a+c) |^ 3 - a^3) + 1/2*c*((b/c)*(a+c))*(a+c+a)
 by Qq,Dd
 .= (1/3*(b/c))* ( (a^3 - (a-c)^2*(a-c) ) +((-1)*((a+c) |^ 3 - a^3)))
       + 1/2*c*(b/c)*( (a+c)*(2*a+c)  +(c-a)*(2*a-c) )
 .= (1/3*(b/c))* ( (a^3 - (((a ^2) - ((2 * a) * c)) + (c ^2))*(a-c) )
 +((-1)*((a+c)^2*(a+c) - a^3)))
 + 1/2*c*(b/c)*( (a*(2*a)+a*c+c*(2*a)+c*c) +(c*(2*a)-c*c-a*(2*a)+a*c) )
 by POLYEQ_3:27
 .= (1/3*(b/c))*(
 a^3 - (a^2*a-a^2*c - 2*a*c*a+2*a*c*c + c^2*a-c^2*c)
+ (-1)*( (a^2*a+a^2*c + 2*a*c*a+2*a*c*c + c^2*a+c^2*c) - a^3 )
 ) + 1/2*(c*(b/c))*( 6*a*c )
 .= (1/3*(b/c))*(
 a^3 - (a^2*a-a^2*c - 2*a*c*a+2*a*c*c + c^2*a-c^2*c)
+ (-1)*( (a^2*a+a^2*c + 2*a*c*a+2*a*c*c + c^2*a+c^2*c) - a^3 )
 ) + 1/2*(c*b/c)*( 6*a*c ) by XCMPLX_1:74
 .= (1/3*(b/c))*(
 a^3 - a^2*a+a^2*c + 2*a*c*a - 2*a*c*c - c^2*a+c^2*c
+ ( (-1)*a^2*a+ (-1)*a^2*c + (-1)*2*a*c*a+(-1)*2*a*c*c +
(-1)*c^2*a+(-1)*c^2*c - (-1)*a^3 )
 ) + 1/2*(b)*( 6*a*c ) by XCMPLX_1:89,A2
 .= (1/3*(b/c))*( a^3 - a^2*a + a*a*c  - 4*a*c*c - c^2*a  -a^2*a+ -a^2*c
-c^2*a+ - -a^3) + 1/2*6*b*a*c
 .= (1/3*(b/c))*(
  a^3 - a^2*a + a*a*c - 4*a*c*c - c^2*a  -a^3+ -a^2*c
     -c^2*a+ - -a^3) + 1/2*6*b*a*c by POLYEQ_3:def 3
  .= (1/3*(b/c))*(-a^2*a +a*a*c -4*a*c*c -c^2*a -a^2*c -c^2*a +a^3
 ) + 3*b*a*c
  .= (1/3*(b/c))*(-a^3 +a*a*c -4*a*c*c -c^2*a -a^2*c -c^2*a +a^3
 ) + 3*b*a*c by POLYEQ_3:def 3
 .= (-6)*(1/3)*((b/c)*c)* a*c + 3*b*a*c
 .= (-6)*(1/3)*((c*b/c))* a*c + 3*b*a*c by XCMPLX_1:74
 .= (-6)*(1/3)*(b)* a*c + 3*b*a*c by XCMPLX_1:89,A2
 .= a*(b*c);
  1/2*s*((a-(a-c))*(a+(a-c))) + d*c + 1/2*p*(((a+c)-a)*((a+c)+a)) + q*c
 = 1/2*((b/c)*c)*(2*a-c) - 1/2*((b/c)*c)*(2*a+c)
  + ((b/c)*c)*(c-a)+((b/c)*c)*(c+a) by Dd,Qq
 .= 1/2*((b/c)*c)*(2*a-c) - 1/2*((b/c)*c)*(2*a+c)
  + ((b*c/c))*(c-a)+((b/c)*c)*(c+a) by XCMPLX_1:74
 .= 1/2*((b/c)*c)*(2*a-c) - 1/2*((b/c)*c)*(2*a+c)
  + ((b))*(c-a)+((b/c)*c)*(c+a) by XCMPLX_1:89,A2
 .= 1/2*((b*c/c))*(2*a-c) - 1/2*((b/c)*c)*(2*a+c)
  + b*(c-a)+((b/c)*c)*(c+a) by XCMPLX_1:74
 .= 1/2*((b*c/c))*(2*a-c) - 1/2*((b*c/c))*(2*a+c)
  + b*(c-a)+((b/c)*c)*(c+a) by XCMPLX_1:74
 .= 1/2*((b*c/c))*(2*a-c) - 1/2*((b*c/c))*(2*a+c)
  + b*(c-a)+((b*c/c))*(c+a) by XCMPLX_1:74
 .= 1/2*((b*c/c))*(2*a-c) - 1/2*((b*c/c))*(2*a+c)
  + b*(c-a)+b*(c+a) by XCMPLX_1:89,A2
 .= 1/2*((b*c/c))*(2*a-c) - 1/2*b*(2*a+c)
  + b*(c-a)+b*(c+a) by XCMPLX_1:89,A2
 .= 1/2*b*(2*a-c) - 1/2*b*(2*a+c) + b*(c-a)+b*(c+a) by XCMPLX_1:89,A2
 .= b*c;
 hence centroid(f,C) = a by XCMPLX_1:89,A1,A2,CE,B1;
end;
