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
  x0 in dom tan & x1 in dom tan implies
  [!tan(#)sin,x0,x1!] = (1/cos(x0)-cos(x0)-1/cos(x1)+cos(x1))/(x0-x1)
proof
  assume
A1:x0 in dom tan & x1 in dom tan;
  [!tan(#)sin,x0,x1!] = ((tan.x0)*(sin.x0)-(tan(#)sin).x1)
       /(x0-x1) by VALUED_1:5
    .= ((tan.x0*sin.x0)-(tan.x1*sin.x1))/(x0-x1) by VALUED_1:5
    .= (((sin.x0*(cos.x0)")*sin.x0)-(tan.x1*sin.x1))/(x0-x1)
                                                   by A1,RFUNCT_1:def 1
    .= (sin(x0)/cos(x0)*sin(x0)-sin(x1)/cos(x1)*sin(x1))/(x0-x1)
                                                   by A1,RFUNCT_1:def 1
    .= (sin(x0)/(cos(x0)/sin(x0))-sin(x1)/cos(x1)*sin(x1))/(x0-x1)
                                                           by XCMPLX_1:82
    .= (sin(x0)/(cos(x0)/sin(x0))-sin(x1)/(cos(x1)/sin(x1)))/(x0-x1)
                                                           by XCMPLX_1:82
    .= ((sin(x0)*sin(x0))/cos(x0)-sin(x1)/(cos(x1)/sin(x1)))/(x0-x1)
                                                       by XCMPLX_1:77
    .= ((sin(x0)*sin(x0))/cos(x0)-(sin(x1)*sin(x1))/cos(x1))/(x0-x1)
                                                       by XCMPLX_1:77
    .= ((1-cos(x0)*cos(x0))/cos(x0)-(sin(x1)*sin(x1))/cos(x1))/(x0-x1)
                                                       by SIN_COS4:4
    .= ((1/cos(x0)-(cos(x0)*cos(x0)/cos(x0))
       -(1-cos(x1)*cos(x1))/cos(x1)))/(x0-x1) by SIN_COS4:4
    .= ((1/cos(x0)-cos(x0))
       -(1/cos(x1)-cos(x1)*cos(x1)/cos(x1)))/(x0-x1)
                                         by A1,FDIFF_8:1,XCMPLX_1:89
    .= ((1/cos(x0)-cos(x0))-(1/cos(x1)-cos(x1)))/(x0-x1)
                                        by A1,FDIFF_8:1,XCMPLX_1:89
    .= (1/cos(x0)-cos(x0)-1/cos(x1)+cos(x1))/(x0-x1);
  hence thesis;
end;
