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(x+h/2)<>0 & sin(x-h/2)<>0
  implies cD(f,h).x = -2*(sin(x-h/2)-sin(x+h/2))/(cos(2*x)-cos(h))
proof
  assume that
A1:for x holds f.x = 1/sin(x) and
A2:sin(x+h/2)<>0 & sin(x-h/2)<>0;
  cD(f,h).x = f.(x+h/2) - f.(x-h/2) by DIFF_1:5
    .= 1/sin(x+h/2) - f.(x-h/2) by A1
    .= 1/sin(x+h/2) - 1/sin(x-h/2) by A1
    .= (1*sin(x-h/2)-1*sin(x+h/2))/(sin(x+h/2)*sin(x-h/2)) by A2,XCMPLX_1:130
    .= (sin(x-h/2)-sin(x+h/2))
       /(-(1/2)*(cos((x+h/2)+(x-h/2))-cos((x+h/2)-(x-h/2)))) by SIN_COS4:29
    .= (sin(x-h/2)-sin(x+h/2))/((-1/2)*(cos(2*x)-cos(h)))
    .= (sin(x-h/2)-sin(x+h/2))/(-1/2)/(cos(2*x)-cos(h)) by XCMPLX_1:78
    .= (-2)*((sin(x-h/2)-sin(x+h/2))/(cos(2*x)-cos(h)));
  hence thesis;
end;
