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 = (tan(#)sin).x) & x in dom tan & x+h in dom tan
  implies fD(f,h).x = 1/cos(x+h)-cos(x+h)-1/cos(x)+cos(x)
proof
  assume that
A1:for x holds f.x = (tan(#)sin).x and
A2:x in dom tan & x+h in dom tan;
  fD(f,h).x = f.(x+h) - f.x by DIFF_1:3
    .= (tan(#)sin).(x+h) - f.x by A1
    .= (tan(#)sin).(x+h) - (tan(#)sin).x by A1
    .= (tan.(x+h))*(sin.(x+h)) - (tan(#)sin).x by VALUED_1:5
    .= (tan.(x+h))*(sin.(x+h)) - (tan.(x))*(sin.(x)) by VALUED_1:5
    .= (sin.(x+h)*(cos.(x+h))")*(sin.(x+h)) - (tan.(x))*(sin.(x))
                                                     by A2,RFUNCT_1:def 1
    .= sin(x+h)/cos(x+h)*sin(x+h)-sin(x)/cos(x)*sin(x) by A2,RFUNCT_1:def 1
    .= sin(x+h)/(cos(x+h)/sin(x+h))-sin(x)/cos(x)*sin(x) by XCMPLX_1:82
    .= sin(x+h)/(cos(x+h)/sin(x+h))-sin(x)/(cos(x)/sin(x)) by XCMPLX_1:82
    .= (sin(x+h)*sin(x+h))/cos(x+h)-sin(x)/(cos(x)/sin(x)) by XCMPLX_1:77
    .= (sin(x+h)*sin(x+h))/cos(x+h)-(sin(x)*sin(x))/cos(x) by XCMPLX_1:77
    .= (1-cos(x+h)*cos(x+h))/cos(x+h)-(sin(x)*sin(x))/cos(x) by SIN_COS4:4
    .= (1/cos(x+h)-(cos(x+h)*cos(x+h))/cos(x+h))
       -(1-cos(x)*cos(x))/cos(x) by SIN_COS4:4
    .= (1/cos(x+h)-cos(x+h))
       -(1/cos(x)-(cos(x)*cos(x))/cos(x)) by A2,FDIFF_8:1,XCMPLX_1:89
    .= (1/cos(x+h)-cos(x+h))-(1/cos(x)-cos(x)) by A2,FDIFF_8:1,XCMPLX_1:89
    .= 1/cos(x+h)-cos(x+h)-1/cos(x)+cos(x);
  hence thesis;
end;
