reserve r,x,x0,a,b for Real;
reserve n,m for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve Z for open Subset of REAL;

theorem
  A = [.0,PI.] implies integral(sin^2,A) = PI/2
proof
  assume A=[.0,PI.];
  then upper_bound A=PI & lower_bound A=0 by INTEGRA8:37;
  then
  integral(sin^2,A)
     =(AffineMap(1/2,0)-(1/4)(#)(sin*AffineMap(2,0))).pp -
  (AffineMap(1/2,0)-(1/4)(#)(sin*AffineMap(2,0))).0 by Th13
    .= AffineMap(1/2,0).(PI)-((1/4)(#)(sin*AffineMap(2,0))).(PI) -(AffineMap
  (1/2,0)-(1/4)(#)(sin*AffineMap(2,0))).0 by Lm5,VALUED_1:13
    .= AffineMap(1/2,0).(PI)-((1/4)(#)(sin*AffineMap(2,0))).(PI) -(AffineMap
  (1/2,0).0-((1/4)(#)(sin*AffineMap(2,0))).0) by Lm5,VALUED_1:13,Lm6
    .= (1/2)*(PI)+0 -((1/4)(#)(sin*AffineMap(2,0))).(PI) -(AffineMap(1/2,0).
  0-((1/4)(#)(sin*AffineMap(2,0))).0) by FCONT_1:def 4
    .= (1/2)*(PI)-(1/4)*(sin*AffineMap(2,0)).pp -(AffineMap(1/2,0).0-((1/4
  )(#)(sin*AffineMap(2,0))).0) by VALUED_1:6
    .= (1/2)*(PI)-(1/4)*(sin.(AffineMap(2,0).(PI))) -(AffineMap(1/2,0).0-((1
  /4)(#)(sin*AffineMap(2,0))).0) by Lm4,FUNCT_1:13
    .= (1/2)*(PI)-(1/4)*(sin.(2*PI+0)) -(AffineMap(1/2,0).0-((1/4)(#)(sin*
  AffineMap(2,0))).0) by FCONT_1:def 4
    .= (1/2)*(PI)-(1/4)*sin.(2*PI+0) -(0-((1/4)(#)(sin*AffineMap(2,0))).0)
  by FCONT_1:48
    .= (1/2)*(PI)-(1/4)*sin.(2*PI+0) -0+((1/4)(#)(sin*AffineMap(2,0))).0
    .= (1/2)*(PI)-(1/4)*sin.(2*PI+0) +(1/4)*(sin*AffineMap(2,0)).0 by
VALUED_1:6
    .= (1/2)*(PI)-(1/4)*sin.(2*PI+0) +(1/4)*(sin.(AffineMap(2,0).0)) by Lm4,
FUNCT_1:13,Lm6
    .= (1/2)*(PI)-(1/4)*sin.(0+2*PI*1)+(1/4)*sin.0 by FCONT_1:48
    .= (1/2)*(PI)-(1/4)*sin.0+(1/4)*sin.0 by SIN_COS6:8
    .= PI/2;
  hence thesis;
end;
