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=1/(cos(x))^2) & x0<>x1 & cos(x0)<>0 & cos(x1)<>0
  implies [!f,x0,x1!] = (-16)*sin((x1+x0)/2)*sin((x1-x0)/2)*cos((x1+x0)/2)
  *cos((x1-x0)/2)/((cos(x0+x1)+cos(x0-x1))^2)/(x0-x1)
proof
  assume that
A1:for x holds f.x=1/(cos(x))^2 and
  x0<>x1 and
A2:cos(x0)<>0 & cos(x1)<>0;
f.x0=1/(cos(x0))^2 & f.x1=1/(cos(x1))^2 by A1;
  then [!f,x0,x1!] = ((1*(cos(x1))^2-1*(cos(x0))^2)/((cos(x0))^2*(cos(x1))^2))
       /(x0-x1) by A2,XCMPLX_1:130
    .= (((cos(x1))^2-(cos(x0))^2)/((cos(x0)*cos(x1))^2))/(x0-x1)
    .= ((cos(x1))^2-(cos(x0))^2)/(((1/2)*(cos(x0+x1)+cos(x0-x1)))^2)/(x0-x1)
                                                              by SIN_COS4:32
    .= ((cos(x1))^2-(cos(x0))^2)/((1/4)*(cos(x0+x1)+cos(x0-x1))^2)/(x0-x1)
    .= ((cos(x1))^2-(cos(x0))^2)/(1/4)/((cos(x0+x1)+cos(x0-x1))^2)/(x0-x1)
                                                              by XCMPLX_1:78
    .= 4*((cos(x1)-cos(x0))*(cos(x1)+cos(x0)))
       /((cos(x0+x1)+cos(x0-x1))^2)/(x0-x1)
    .= 4*((-2*(sin((x1+x0)/2)*sin((x1-x0)/2)))*(cos(x1)+cos(x0)))
       /((cos(x0+x1)+cos(x0-x1))^2)/(x0-x1) by SIN_COS4:18
    .= 4*((-2*(sin((x1+x0)/2)*sin((x1-x0)/2)))
       *(2*(cos((x1+x0)/2)*cos((x1-x0)/2))))
       /((cos(x0+x1)+cos(x0-x1))^2)/(x0-x1) by SIN_COS4:17
    .= (-16)*sin((x1+x0)/2)*sin((x1-x0)/2)*cos((x1+x0)/2)*cos((x1-x0)/2)
       /((cos(x0+x1)+cos(x0-x1))^2)/(x0-x1);
  hence thesis;
end;
