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 = (cot(#)cos).x) & x0 in dom cot & x1 in dom cot
  implies [!f,x0,x1!] = (1/sin(x0)-sin(x0)-1/sin(x1)+sin(x1))/(x0-x1)
proof
  assume that
A1:for x holds f.x = (cot(#)cos).x and
A2:x0 in dom cot & x1 in dom cot;
A3:f.(x0) = (cot(#)cos).x0 by A1;
f.(x1) = (cot(#)cos).x1 by A1;
  then [!f,x0,x1!] = ((cot.(x0))*(cos.(x0)) - (cot(#)cos).x1)/(x0-x1)
                                                 by A3,VALUED_1:5
    .= ((cot.(x0))*(cos.(x0))-(cot.(x1))*(cos.(x1)))/(x0-x1) by VALUED_1:5
    .= ((cos.(x0)*(sin.(x0))")*(cos.(x0))-(cot.(x1))*(cos.(x1)))/(x0-x1)
                                                     by A2,RFUNCT_1:def 1
    .= (cos(x0)/sin(x0)*cos(x0)-cos(x1)/sin(x1)*cos(x1))/(x0-x1)
                                                  by A2,RFUNCT_1:def 1
    .= (cos(x0)/(sin(x0)/cos(x0))-cos(x1)/sin(x1)*cos(x1))/(x0-x1)
                                                            by XCMPLX_1:82
    .= (cos(x0)/(sin(x0)/cos(x0))-cos(x1)/(sin(x1)/cos(x1)))/(x0-x1)
                                                            by XCMPLX_1:82
    .= ((cos(x0)*cos(x0))/sin(x0)-cos(x1)/(sin(x1)/cos(x1)))/(x0-x1)
                                                       by XCMPLX_1:77
    .= ((cos(x0)*cos(x0))/sin(x0)-(cos(x1)*cos(x1))/sin(x1))/(x0-x1)
                                                       by XCMPLX_1:77
    .= ((1-sin(x0)*sin(x0))/sin(x0)-(cos(x1)*cos(x1))/sin(x1))/(x0-x1)
                                                       by SIN_COS4:5
    .= ((1/sin(x0)-sin(x0)*sin(x0)/sin(x0))
       -(1-sin(x1)*sin(x1))/sin(x1))/(x0-x1) by SIN_COS4:5
    .= ((1/sin(x0)-sin(x0))
       -(1/sin(x1)-sin(x1)*sin(x1)/sin(x1)))/(x0-x1)
                                              by A2,FDIFF_8:2,XCMPLX_1:89
    .= ((1/sin(x0)-sin(x0))-(1/sin(x1)-sin(x1)))/(x0-x1)
                                              by A2,FDIFF_8:2,XCMPLX_1:89
    .= (1/sin(x0)-sin(x0)-1/sin(x1)+sin(x1))/(x0-x1);
  hence thesis;
end;
