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))^2) & x0<>x1 & sin(x0)<>0 & sin(x1)<>0
  implies [!f,x0,x1!] = 16*cos((x1+x0)/2)*sin((x1-x0)/2)
  *cos((x1-x0)/2)*sin((x1+x0)/2)/(((cos(x0+x1)-cos(x0-x1))^2)*(x0-x1))
proof
  assume that
A1:for x holds f.x=1/(sin(x))^2 and
  x0<>x1 and
A2:sin(x0)<>0 & sin(x1)<>0;
f.x0=1/(sin(x0))^2 & f.x1=1/(sin(x1))^2 by A1;
  then [!f,x0,x1!] = ((1*(sin(x1))^2-1*(sin(x0))^2)/((sin(x0))^2*(sin(x1))^2))
       /(x0-x1) by A2,XCMPLX_1:130
    .= (((sin(x1))^2-(sin(x0))^2)/((sin(x0)*sin(x1))^2))/(x0-x1)
    .= ((sin(x1))^2-(sin(x0))^2)/((-(1/2)*(cos(x0+x1)-cos(x0-x1)))^2)
       /(x0-x1) by SIN_COS4:29
    .= ((sin(x1))^2-(sin(x0))^2)/((1/4)*(cos(x0+x1)-cos(x0-x1))^2)
       /(x0-x1)
    .= ((sin(x1))^2-(sin(x0))^2)/(1/4)/((cos(x0+x1)-cos(x0-x1))^2)
       /(x0-x1) by XCMPLX_1:78
    .= 4*(((sin(x1))-(sin(x0)))*((sin(x1))+(sin(x0))))
       /((cos(x0+x1)-cos(x0-x1))^2)/(x0-x1)
    .= 4*((2*(cos((x1+x0)/2)*sin((x1-x0)/2)))*(sin(x1)+sin(x0)))
       /((cos(x0+x1)-cos(x0-x1))^2)/(x0-x1) by SIN_COS4:16
    .= 4*((2*(cos((x1+x0)/2)*sin((x1-x0)/2)))
       *(2*(cos((x1-x0)/2)*sin((x1+x0)/2))))
       /((cos(x0+x1)-cos(x0-x1))^2)/(x0-x1) by SIN_COS4:15
    .= 16*cos((x1+x0)/2)*sin((x1-x0)/2)*cos((x1-x0)/2)*sin((x1+x0)/2)
       /(((cos(x0+x1)-cos(x0-x1))^2)*(x0-x1)) by XCMPLX_1:78;
  hence thesis;
end;
