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