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 cosec & x1 in dom cosec implies [!cosec,x0,x1!] = 2*cos((x1+
  x0)/2)*sin((x1-x0)/2) /(sin(x1)*sin(x0)*(x0-x1))
proof
  assume that
A1: x0 in dom cosec and
A2: x1 in dom cosec;
A3: cosec.x1 = (sin.x1)" by A2,RFUNCT_1:def 2
    .= cosec(x1) by XCMPLX_1:215;
  cosec.x0 = (sin.x0)" by A1,RFUNCT_1:def 2
    .= cosec(x0) by XCMPLX_1:215;
  then
  [!cosec,x0,x1!] = (1*sin(x1)/(sin(x0)*sin(x1)) -1/sin(x1))/(x0-x1) by A2,A3,
RFUNCT_1:3,XCMPLX_1:91
    .= (1*sin(x1)/(sin(x0)*sin(x1)) -1*sin(x0)/(sin(x1)*sin(x0)))/(x0-x1) by A1
,RFUNCT_1:3,XCMPLX_1:91
    .= (sin(x1)-sin(x0))/(sin(x1)*sin(x0))/(x0-x1) by XCMPLX_1:120
    .= (sin(x1)-sin(x0))/(sin(x1)*sin(x0)*(x0-x1)) by XCMPLX_1:78
    .= 2*(cos((x1+x0)/2)*sin((x1-x0)/2)) /(sin(x1)*sin(x0)*(x0-x1)) by
SIN_COS4:16;
  hence thesis;
end;
