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 bD(f,h).x = 1/sin(x)-sin(x)-1/sin(x-h)+sin(x-h)
proof
  assume that
A1:for x holds f.x = (cot(#)cos).x and
A2:x in dom cot & x-h in dom cot;
  bD(f,h).x = f.x - f.(x-h) by DIFF_1:4
    .= (cot(#)cos).x - f.(x-h) by A1
    .= (cot(#)cos).x - (cot(#)cos).(x-h) by A1
    .= (cot.(x))*(cos.(x)) - (cot(#)cos).(x-h) by VALUED_1:5
    .= (cot.(x))*(cos.(x)) - (cot.(x-h))*(cos.(x-h)) by VALUED_1:5
    .= (cos.(x)*(sin.(x))")*(cos.(x))-(cot.(x-h))*(cos.(x-h))
                                                  by A2,RFUNCT_1:def 1
    .= cos(x)/sin(x)*cos(x)-cos(x-h)/sin(x-h)*cos(x-h) by A2,RFUNCT_1:def 1
    .= cos(x)/(sin(x)/cos(x))-cos(x-h)/sin(x-h)*cos(x-h) by XCMPLX_1:82
    .= cos(x)/(sin(x)/cos(x))-cos(x-h)/(sin(x-h)/cos(x-h)) by XCMPLX_1:82
    .= (cos(x)*cos(x))/sin(x)-cos(x-h)/(sin(x-h)/cos(x-h))
                                                       by XCMPLX_1:77
    .= (cos(x)*cos(x))/sin(x)-(cos(x-h)*cos(x-h))/sin(x-h)
                                                       by XCMPLX_1:77
    .= (1-sin(x)*sin(x))/sin(x)-(cos(x-h)*cos(x-h))/sin(x-h)
                                                       by SIN_COS4:5
    .= (1/sin(x)-sin(x)*sin(x)/sin(x))
       -(1-sin(x-h)*sin(x-h))/sin(x-h) by SIN_COS4:5
    .= (1/sin(x)-sin(x))
       -(1/sin(x-h)-sin(x-h)*sin(x-h)/sin(x-h)) by A2,FDIFF_8:2,XCMPLX_1:89
    .= (1/sin(x)-sin(x))-(1/sin(x-h)-sin(x-h)) by A2,FDIFF_8:2,XCMPLX_1:89
    .= 1/sin(x)-sin(x)-1/sin(x-h)+sin(x-h);
  hence thesis;
end;
