reserve r,p,x for Real;
reserve n for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve Z for open Subset of REAL;

theorem
  n<>0 implies integral(sin*(AffineMap(n,0)),A) = ((-1/n)(#)(cos*(
  AffineMap(n,0)))).(upper_bound A) -((-1/n)(#)(cos*(AffineMap(n,0)))).(
  lower_bound A)
proof
  assume
A1: n<>0;
A2: [#]REAL = dom (sin*(AffineMap(n,0))) by FUNCT_2:def 1;
A3: for x st x in REAL holds AffineMap(n,0).x=n*x
  proof
    let x;
    assume x in REAL;
    (AffineMap(n,0)).x = n*x + 0 by FCONT_1:def 4
      .=n*x;
    hence thesis;
  end;
A4: for x being Element of REAL
  st x in dom (((-1/n)(#)(cos*(AffineMap(n,0))))`|REAL) holds (((-1/
  n)(#)(cos*(AffineMap(n,0))))`|REAL).x = (sin*(AffineMap(n,0))).x
  proof
    let x be Element of REAL;
    assume
   x in dom (((-1/n)(#)(cos*(AffineMap(n,0))))`|REAL);
    (((-1/n)(#)(cos*(AffineMap(n,0))))`|REAL).x = sin(n*x) by A1,Th4
      .= sin.((AffineMap(n,0)).x) by A3
      .= (sin*(AffineMap(n,0))).x by A2,FUNCT_1:12;
    hence thesis;
  end;
  (-1/n)(#)(cos*(AffineMap(n,0))) is_differentiable_on REAL by A1,Th4;
  then
  dom (((-1/n)(#)(cos*(AffineMap(n,0))))`|REAL) = dom (sin*(AffineMap(n,0
  ))) by A2,FDIFF_1:def 7;
  then
A5: (((-1/n)(#)(cos*(AffineMap(n,0))))`|REAL) = sin*(AffineMap(n,0)) by A4,
PARTFUN1:5;
  (sin*AffineMap(n,0))|A is continuous;
  then
A6: (sin*AffineMap(n,0)) is_integrable_on A by A2,INTEGRA5:11;
  (sin*AffineMap(n,0))|A is bounded by A2,INTEGRA5:10;
  hence thesis by A1,A6,A5,Th4,INTEGRA5:13;
end;
