 reserve a,b,r for Real;
 reserve A for non empty set;
 reserve X,x for set;
 reserve f,g,F,G for PartFunc of REAL,REAL;
 reserve n for Element of NAT;

theorem Th8:
  for Z be open Subset of REAL holds
    sin is_differentiable_on Z & sin`|Z = cos|Z
  & cos is_differentiable_on Z & cos`|Z = -sin|Z
proof
  let Z be open Subset of REAL;
  A1: sin is_differentiable_on Z by FDIFF_1:26,SIN_COS:68; then
  A3: dom(sin`|Z) = Z by FDIFF_1:def 7;
  A2: cos is_differentiable_on Z by FDIFF_1:26,SIN_COS:67; then
  A4: dom(cos`|Z) = Z by FDIFF_1:def 7;
  dom cos = REAL by FUNCT_2:def 1; then
  A5: dom(cos|Z) = Z by RELAT_1:62;
  dom sin = REAL by FUNCT_2:def 1; then
  dom(sin|Z) = Z by RELAT_1:62; then
  A6: dom(-(sin|Z)) = Z by VALUED_1:8;
  A7: now
    let t be object;
    assume
    A8: t in dom(sin`|Z); then
    reconsider x = t as Real;
    thus (sin`|Z).t = diff(sin,x) by A1,A3,A8,FDIFF_1:def 7
    .= cos.x by SIN_COS:68
    .= (cos|Z).t by A3,A8,FUNCT_1:49;
  end;

  now
    let t be object;
    assume
    A9: t in dom(cos`|Z);
    then reconsider x=t as Real;
    thus (cos`|Z).t = diff(cos,x) by A2,A4,A9,FDIFF_1:def 7
      .= -sin.x by SIN_COS:67
      .= -(sin|Z).t by A4,A9,FUNCT_1:49
      .= (-(sin|Z)).t by VALUED_1:8;
  end;
  hence thesis by A3,A4,A5,A6,A7,FDIFF_1:26,FUNCT_1:2,SIN_COS:67,68;
end;
