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/sin(x)) & sin(x0)<>0 & sin(x1)<>0 implies [!f,x0,x1!]
  = -2*(sin(x1)-sin(x0))/(cos(x0+x1)-cos(x0-x1))/(x0-x1)
proof
  assume that
A1:for x holds f.x = 1/sin(x) and
A2:sin(x0)<>0 & sin(x1)<>0;
f.x0 = 1/sin(x0) & f.x1 = 1/sin(x1) by A1;
  then [!f,x0,x1!] = ((1*sin(x1)-1*sin(x0))/(sin(x0)*sin(x1)))/(x0-x1)
                                        by A2,XCMPLX_1:130
    .= ((sin(x1)-sin(x0))/(-(1/2)*(cos(x0+x1)-cos(x0-x1))))
       /(x0-x1) by SIN_COS4:29
    .= (sin(x1)-sin(x0))/((-1/2)*(cos(x0+x1)-cos(x0-x1)))
       /(x0-x1)
    .= (sin(x1)-sin(x0))/(-1/2)/(cos(x0+x1)-cos(x0-x1))
       /(x0-x1) by XCMPLX_1:78
    .= (-2)*(sin(x1)-sin(x0))/(cos(x0+x1)-cos(x0-x1))/(x0-x1);
  hence thesis;
end;
