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