reserve f,f1,f2,g for PartFunc of REAL,REAL;
reserve A for non empty closed_interval Subset of REAL;
reserve p,r,x,x0 for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;

theorem
  A = [.0,PI.] implies integral(-sin,A) = -2
proof
  assume A=[.0,PI.];
  then upper_bound A=PI & lower_bound A=0 by Th37;
  then integral(-sin,A) = -1 - cos.0 by Th46,SIN_COS:76
    .= -1 - sin.(PI/2 - 0) by SIN_COS:78
    .= -1 - 1 by SIN_COS:76;
  hence thesis;
end;
