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