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 Th46:
  integral(-sin,A) = cos.(upper_bound A)-cos.(lower_bound A)
proof
A1: for x being Element of REAL st x in dom (cos`|REAL) holds (cos`|REAL).x
  = (-sin).x
  proof
    let x be Element of REAL;
    assume x in dom (cos`|REAL);
    (cos`|REAL).x = diff(cos,x) by FDIFF_1:def 7,SIN_COS:67
      .= -sin.x by SIN_COS:63;
    hence thesis by VALUED_1:8;
  end;
A2: -sin is_integrable_on A & (-sin)|A is bounded by Lm12;
  dom (-sin) = REAL by FUNCT_2:def 1;
  then dom (cos`|REAL) = dom (-sin) by FDIFF_1:def 7,SIN_COS:67;
  then cos`|REAL = -sin by A1,PARTFUN1:5;
  hence thesis by A2,INTEGRA5:13,SIN_COS:67;
end;
