reserve A for non empty closed_interval Subset of REAL;

theorem Th23B:
for a,b,c,d being Real st c <= d holds
integral((id REAL) (#) (AffineMap (a,b)),c,d)
= 1/3*a*(d*d*d - c*c*c) + 1/2*b*(d*d - c*c)
proof
 let a,b,c,d be Real;
 assume A1: c <= d;
 reconsider D = ['c,d'] as non empty closed_interval Subset of REAL;
 reconsider F2 = ( ( #Z 2)) as PartFunc of REAL,REAL;
 reconsider F1 = ( ( #Z 1)) as PartFunc of REAL,REAL;
 DD: ['c,d'] c= REAL;
 B2: ['c,d'] c= dom ( #Z 2) by FUNCT_2:def 1,DD;
 B3: D c= dom (F1) by FUNCT_2:def 1,DD;
 F1 | D is continuous;then
 B4: F1 is_integrable_on D &  F1 | ['c,d'] is bounded
  by INTEGRA5:10,INTEGRA5:11,B3;
 F2 | D is continuous;then
 B5: F2 is_integrable_on ['c,d'] & F2 | ['c,d'] is bounded
 by INTEGRA5:10,INTEGRA5:11,B2;
 A2: ['c,d'] c= dom (a (#) ( F2)) by FUNCT_2:def 1,DD;
 A3: ['c,d'] c= dom (b (#) ( F1)) by FUNCT_2:def 1,DD;
 (a (#) ( F2)) | D is continuous & (b (#)  F1) | D is continuous;then
 A4: (b (#) F1) is_integrable_on ['c,d'] &
 (a (#) F2) is_integrable_on ['c,d'] & (b (#) F1) | ['c,d'] is bounded
 & (a (#) F2) | ['c,d'] is bounded by INTEGRA5:10,INTEGRA5:11,A2,A3;
 [.(lower_bound ['c,d']),(upper_bound ['c,d']).] = ['c,d'] &
 ['c,d'] = [.c,d.] by A1,INTEGRA5:def 3,INTEGRA1:4; then
 LM: lower_bound D = c & d = upper_bound D by INTEGRA1:5;
 integral((id REAL) (#) (AffineMap (a,b)),c,d)
  = integral((a (#) ( #Z 2)) + (b (#) ( #Z 1)),c,d) by Th23C
 .= integral((a (#) F2),c,d) + integral((b (#) F1),c,d)
  by INTEGRA6:12,A1,A2,A3,A4
 .= a * integral(F2,c,d) + integral((b (#) F1),c,d) by INTEGRA6:10,A1,B2,B5
 .= a * integral(F2,c,d) + b * integral(F1,c,d) by INTEGRA6:10,A1,B3,B4
 .= a * integral(F2,D) + b * integral(F1,c,d) by INTEGRA5:def 4,A1
 .= a * integral(F2,D) + b * integral(F1,D) by INTEGRA5:def 4,A1
 .= a * (((1 / (2 + 1)) * ((upper_bound D) |^ (2 + 1)))
         - ((1 / (2 + 1)) * ((lower_bound D) |^ (2 + 1))))
          + b * integral(F1,D) by INTEGRA9:19
 .= a * (1 / 3 * (d |^ (2 + 1)) - (1 / 3 * (c |^ (2 + 1))))
  + b * ((1 / 2) * (d |^ (1 + 1)) - ((1 / 2) * (c |^ (1 + 1))))
       by LM,INTEGRA9:19
 .= a * (1 / 3) * ( d |^ (2 + 1) - c |^ (2 + 1) )
  + b * (1 / 2) * ( d |^ (1 + 1) - c |^ (1 + 1) )
 .= a * (1 / 3) * ( (d |^ 2) * d - c |^ (2 + 1) )
  + b * (1 / 2) * ( d |^ (1 + 1) - c |^ (1 + 1) )  by NEWTON:6
 .= a * (1 / 3) * ( (d |^ ( 1 + 1)) * d -  (c |^ 2)*c )
  + b * (1 / 2) * ( d |^ (1 + 1) -  c |^ (1 + 1) )  by NEWTON:6
 .= a * (1 / 3) * ( (d |^ ( 1 + 1)) * d -  (c |^ 2)*c )
  + b * (1 / 2) * ( d |^ 1 *d -  c |^ (1 + 1) )  by NEWTON:6
 .= a * (1 / 3) * ( (d |^ ( 1 + 1)) * d -  (c |^ 2)*c )
  + b * (1 / 2) * ( d  *d -  c |^ 1 *c )  by NEWTON:6
 .= a * (1 / 3) * ( (d |^ 1 *d ) * d -  (c |^ (1+1))*c )
  + b * (1 / 2) * ( d  *d -  c  *c )  by NEWTON:6
 .= a * (1 / 3) * ( (d *d ) * d -  (c |^ 1*c)*c )
  + b * (1 / 2) * ( d  *d -  c  *c )  by NEWTON:6;
 hence thesis;
end;
