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 cot & x1 in dom cot implies [!cot,x0,x1!] = - sin(x0-x1)/(
  sin(x0)*sin(x1)*(x0-x1))
proof
  assume that
A1: x0 in dom cot and
A2: x1 in dom cot;
A3: cot.x0 = cos.x0*(sin.x0)" by A1,RFUNCT_1:def 1
    .= cos.x0*(1/(sin.x0)) by XCMPLX_1:215
    .= cot(x0) by XCMPLX_1:99;
A4: cot.x1 = cos.x1*(sin.x1)" by A2,RFUNCT_1:def 1
    .= cos.x1*(1/(sin.x1)) by XCMPLX_1:215
    .= cot(x1) by XCMPLX_1:99;
  sin(x0)<>0 & sin(x1)<>0 by A1,A2,FDIFF_8:2;
  then [!cot,x0,x1!] = (-sin(x0-x1)/(sin(x0)*sin(x1))) /(x0-x1) by A3,A4,
SIN_COS4:24
    .= -sin(x0-x1)/(sin(x0)*sin(x1))/(x0-x1) by XCMPLX_1:187;
  hence thesis by XCMPLX_1:78;
end;
