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*2.] implies integral(sin-cos,A) = 0
proof
  assume A = [.0,PI*2.];
  then upper_bound A=PI*2 & lower_bound A=0 by Th37;
  then
  integral(sin-cos,A) = (-cos).(PI*2) - (-cos).0 - (sin.(PI*2) - sin.0) by Th78
    .= -(cos.(PI*2)) - (-cos).0 - (sin.(PI*2) - sin.0) by VALUED_1:8
    .= -(cos.(PI*2)) - (-cos.0) - (sin.(PI*2) - sin.0) by VALUED_1:8
    .= -(cos.(PI*2)) - (-cos.(0+2*PI)) - (sin.(PI*2) - sin.0) by SIN_COS:78
    .= -1+1 - (0 - 0) by SIN_COS:76,78;
  hence thesis;
end;
