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(#)tan).x) & x0 in dom tan & x1 in dom tan
  implies [!f,x0,x1!] = ((cos(x1))^2-(cos(x0))^2)/((cos(x0)*cos(x1))^2*(x0-x1))
proof
  assume that
A1:for x holds f.x=(tan(#)tan).x and
A2:x0 in dom tan & x1 in dom tan;
A3:cos(x0)<>0 & cos(x1)<>0 by A2,FDIFF_8:1;
A4:f.x0=(tan(#)tan).x0 by A1;
f.x1=(tan(#)tan).x1 by A1;
  then [!f,x0,x1!] = (tan.x0*tan.x0-(tan(#)tan).x1)/(x0-x1) by A4,VALUED_1:5
    .= ((tan.x0*tan.x0)-(tan.x1*tan.x1))/(x0-x1) by VALUED_1:5
    .= (((sin.x0*(cos.x0)")*tan.x0)-(tan.x1*tan.x1))/(x0-x1)
                                               by A2,RFUNCT_1:def 1
    .= (((sin.x0*(cos.x0)")*(sin.x0*(cos.x0)"))-(tan.x1*tan.x1))/(x0-x1)
                                                     by A2,RFUNCT_1:def 1
    .= (((sin.x0*(cos.x0)")*(sin.x0*(cos.x0)"))
       -((sin.x1*(cos.x1)")*tan.x1))/(x0-x1) by A2,RFUNCT_1:def 1
    .= ((tan(x0))^2-(tan(x1))^2)/(x0-x1) by A2,RFUNCT_1:def 1
    .= ((tan(x0)-tan(x1))*(tan(x0)+tan(x1)))/(x0-x1)
    .= ((sin(x0-x1)/(cos(x0)*cos(x1)))*(tan(x0)+tan(x1)))/(x0-x1)
                                                            by A3,SIN_COS4:20
    .= ((sin(x0-x1)/(cos(x0)*cos(x1)))*(sin(x0+x1)/(cos(x0)*cos(x1))))/(x0-x1)
                                                             by A3,SIN_COS4:19
    .= ((sin(x0+x1)*sin(x0-x1))/((cos(x0)*cos(x1))^2))/(x0-x1) by XCMPLX_1:76
    .= ((cos(x1))^2-(cos(x0))^2)/((cos(x0)*cos(x1))^2)/(x0-x1) by SIN_COS4:38
    .= ((cos(x1))^2-(cos(x0))^2)/((cos(x0)*cos(x1))^2*(x0-x1)) by XCMPLX_1:78;
  hence thesis;
end;
