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 = [.0,2*PI.] implies integral(cos^2,A) = PI
proof
  assume A=[.0,2*PI.];
  then upper_bound A=2*PI & lower_bound A=0 by INTEGRA8:37;
  then
  integral(cos^2,A)=(AffineMap(1/2,0)+(1/4)(#)(sin*AffineMap(2,0))).(2*PI)
  -(AffineMap(1/2,0)+(1/4)(#)(sin*AffineMap(2,0))).0 by Th16
    .= AffineMap(1/2,0).(2*PI)+((1/4)(#)(sin*AffineMap(2,0))).dp -(
  AffineMap(1/2,0)+(1/4)(#)(sin*AffineMap(2,0))).0 by VALUED_1:1
    .= AffineMap(1/2,0).(2*PI)+((1/4)(#)(sin*AffineMap(2,0))).(2*PI) -(
  AffineMap(1/2,0).0+((1/4)(#)(sin*AffineMap(2,0))).0) by VALUED_1:1,Lm6
    .= (1/2)*(2*PI)+0 +((1/4)(#)(sin*AffineMap(2,0))).dp -(AffineMap(1/2
  ,0).0+((1/4)(#)(sin*AffineMap(2,0))).0) by FCONT_1:def 4
    .= (1/2)*(2*PI)+(1/4)*(sin*AffineMap(2,0)).(2*PI) -(AffineMap(1/2,0).0+(
  (1/4)(#)(sin*AffineMap(2,0))).0) by VALUED_1:6
    .= (1/2)*(2*PI)+(1/4)*(sin.(AffineMap(2,0).dp)) -(AffineMap(1/2,0).0
  +((1/4)(#)(sin*AffineMap(2,0))).0) by Lm4,FUNCT_1:13
    .= (1/2)*(2*PI)+(1/4)*(sin.ddp) -(AffineMap(1/2,0).0+((1/4)(#)(
  sin*AffineMap(2,0))).0) by FCONT_1:def 4
    .= (1/2)*(2*PI)+(1/4)*sin.(2*2*PI+0) -(0+((1/4)(#)(sin*AffineMap(2,0))).
  0) by FCONT_1:48
    .= (1/2)*(2*PI)+(1/4)*sin.(2*2*PI+0) -(1/4)*(sin*AffineMap(2,0)).0 by
VALUED_1:6
    .= (1/2)*(2*PI)+(1/4)*sin.(2*2*PI+0) -(1/4)*(sin.(AffineMap(2,0).0)) by Lm4
,FUNCT_1:13,Lm6
    .= (1/2)*(2*PI)+(1/4)*sin.(0+2*PI*2)-(1/4)*sin.0 by FCONT_1:48
    .= (1/2)*(2*PI)+(1/4)*sin.0-(1/4)*sin.0 by SIN_COS6:8
    .= PI;
  hence thesis;
end;
