reserve n,m for Element of NAT;
reserve h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
reserve f,f1,f2 for Function of REAL,REAL;
reserve S for Seq_Sequence;

theorem
  (for x holds f.x = (cot(#)cos).x) & x in dom cot & x+h in dom cot
  implies fD(f,h).x = 1/sin(x+h)-sin(x+h)-1/sin(x)+sin(x)
proof
  assume that
A1:for x holds f.x = (cot(#)cos).x and
A2:x in dom cot & x+h in dom cot;
  fD(f,h).x = f.(x+h) - f.x by DIFF_1:3
    .= (cot(#)cos).(x+h) - f.x by A1
    .= (cot(#)cos).(x+h) - (cot(#)cos).x by A1
    .= (cot.(x+h))*(cos.(x+h)) - (cot(#)cos).x by VALUED_1:5
    .= (cot.(x+h))*(cos.(x+h)) - (cot.(x))*(cos.(x)) by VALUED_1:5
    .= (cos.(x+h)*(sin.(x+h))")*(cos.(x+h))-(cot.(x))*(cos.(x))
                                                by A2,RFUNCT_1:def 1
    .= cos(x+h)/sin(x+h)*cos(x+h)-cos(x)/sin(x)*cos(x) by A2,RFUNCT_1:def 1
    .= cos(x+h)/(sin(x+h)/cos(x+h))-cos(x)/sin(x)*cos(x) by XCMPLX_1:82
    .= cos(x+h)/(sin(x+h)/cos(x+h))-cos(x)/(sin(x)/cos(x)) by XCMPLX_1:82
    .= (cos(x+h)*cos(x+h))/sin(x+h)-cos(x)/(sin(x)/cos(x))
                                                       by XCMPLX_1:77
    .= (cos(x+h)*cos(x+h))/sin(x+h)-(cos(x)*cos(x))/sin(x)
                                                       by XCMPLX_1:77
    .= (1-sin(x+h)*sin(x+h))/sin(x+h)-(cos(x)*cos(x))/sin(x)
                                                       by SIN_COS4:5
    .= (1/sin(x+h)-sin(x+h)*sin(x+h)/sin(x+h))
       -(1-sin(x)*sin(x))/sin(x) by SIN_COS4:5
    .= (1/sin(x+h)-sin(x+h))
       -(1/sin(x)-sin(x)*sin(x)/sin(x)) by A2,FDIFF_8:2,XCMPLX_1:89
    .= (1/sin(x+h)-sin(x+h))-(1/sin(x)-sin(x)) by A2,FDIFF_8:2,XCMPLX_1:89
    .= 1/sin(x+h)-sin(x+h)-1/sin(x)+sin(x);
  hence thesis;
end;
