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(#)cos).x) & x in dom tan & x-h in dom tan
  implies bD(f,h).x = sin(x)-sin(x-h)
proof
  assume that
A1:for x holds f.x = (tan(#)cos).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(#)cos).x - f.(x-h) by A1
    .= (tan(#)cos).x - (tan(#)cos).(x-h) by A1
    .= (tan.(x))*(cos.(x)) - (tan(#)cos).(x-h) by VALUED_1:5
    .= (tan.(x))*(cos.(x)) - (tan.(x-h))*(cos.(x-h)) by VALUED_1:5
    .= (sin.(x)*(cos.(x))")*(cos.(x))-(tan.(x-h))*(cos.(x-h))
                                                     by A2,RFUNCT_1:def 1
    .= sin(x)/cos(x)*cos(x)-sin(x-h)/cos(x-h)*cos(x-h) by A2,RFUNCT_1:def 1
    .= sin(x)/(cos(x)/cos(x))-sin(x-h)/cos(x-h)*cos(x-h) by XCMPLX_1:82
    .= sin(x)/(cos(x)*(1/cos(x)))-sin(x-h)/(cos(x-h)/cos(x-h)) by XCMPLX_1:82
    .= sin(x)/1-sin(x-h)/(cos(x-h)*(1/cos(x-h))) by A2,FDIFF_8:1,XCMPLX_1:106
    .= sin(x)/1-sin(x-h)/1 by A2,FDIFF_8:1,XCMPLX_1:106
    .= sin(x)-sin(x-h);
  hence thesis;
end;
