reserve h,r,r1,r2,x0,x1,x2,x3,x4,x5,x,a,b,c,k for Real,
  f,f1,f2 for Function of REAL,REAL;

theorem
  x0 in dom sec & x1 in dom sec implies [!sec,x0,x1!] = -2*sin((x1+x0)/2
  ) *sin((x1-x0)/2)/(cos(x1)*cos(x0)*(x0-x1))
proof
  assume that
A1: x0 in dom sec and
A2: x1 in dom sec;
A3: sec.x1 = (cos.x1)" by A2,RFUNCT_1:def 2
    .= sec(x1) by XCMPLX_1:215;
  sec.x0 = (cos.x0)" by A1,RFUNCT_1:def 2
    .= sec(x0) by XCMPLX_1:215;
  then
  [!sec,x0,x1!] = (1*cos(x1)/(cos(x0)*cos(x1)) -1/cos(x1))/(x0-x1) by A2,A3,
RFUNCT_1:3,XCMPLX_1:91
    .= (1*cos(x1)/(cos(x0)*cos(x1)) -1*cos(x0)/(cos(x1)*cos(x0)))/(x0-x1) by A1
,RFUNCT_1:3,XCMPLX_1:91
    .= (cos(x1)-cos(x0))/(cos(x1)*cos(x0))/(x0-x1) by XCMPLX_1:120
    .= (cos(x1)-cos(x0))/(cos(x1)*cos(x0)*(x0-x1)) by XCMPLX_1:78
    .= (-2*(sin((x1+x0)/2)*sin((x1-x0)/2))) /(cos(x1)*cos(x0)*(x0-x1)) by
SIN_COS4:18
    .= -2*(sin((x1+x0)/2)*sin((x1-x0)/2)) /(cos(x1)*cos(x0)*(x0-x1)) by
XCMPLX_1:187;
  hence thesis;
end;
