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+h/2 in dom tan & x-h/2 in dom tan
  implies cD(f,h).x = sin(x+h/2)-sin(x-h/2)
proof
  assume that
A1:for x holds f.x = (tan(#)cos).x and
A2:x+h/2 in dom tan & x-h/2 in dom tan;
  cD(f,h).x = f.(x+h/2) - f.(x-h/2) by DIFF_1:5
    .= (tan(#)cos).(x+h/2) - f.(x-h/2) by A1
    .= (tan(#)cos).(x+h/2) - (tan(#)cos).(x-h/2) by A1
    .= (tan.(x+h/2))*(cos.(x+h/2)) - (tan(#)cos).(x-h/2) by VALUED_1:5
    .= (tan.(x+h/2))*(cos.(x+h/2)) - (tan.(x-h/2))*(cos.(x-h/2)) by VALUED_1:5
    .= (sin.(x+h/2)*(cos.(x+h/2))")*(cos.(x+h/2))-(tan.(x-h/2))*(cos.(x-h/2))
                                                     by A2,RFUNCT_1:def 1
    .= sin(x+h/2)/cos(x+h/2)*cos(x+h/2)-sin(x-h/2)/cos(x-h/2)*cos(x-h/2)
                                                       by A2,RFUNCT_1:def 1
    .= sin(x+h/2)/(cos(x+h/2)/cos(x+h/2))-sin(x-h/2)/cos(x-h/2)*cos(x-h/2)
                                                               by XCMPLX_1:82
    .= sin(x+h/2)/(cos(x+h/2)*(1/cos(x+h/2)))
       -sin(x-h/2)/(cos(x-h/2)/cos(x-h/2)) by XCMPLX_1:82
    .= sin(x+h/2)/1-sin(x-h/2)/(cos(x-h/2)*(1/cos(x-h/2)))
                                               by A2,FDIFF_8:1,XCMPLX_1:106
    .= sin(x+h/2)/1-sin(x-h/2)/1 by A2,FDIFF_8:1,XCMPLX_1:106
    .= sin(x+h/2)-sin(x-h/2);
  hence thesis;
end;
