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/cos(x)) & cos(x)<>0 & cos(x+h)<>0
  implies fD(f,h).x = 2*(cos(x)-cos(x+h))/(cos(2*x+h)+cos(h))
proof
  assume that
A1:for x holds f.x = 1/cos(x) and
A2:cos(x)<>0 & cos(x+h)<>0;
f.(x+h) = 1/cos(x+h) by A1;
  then fD(f,h).x = 1/cos(x+h) - f.x by DIFF_1:3
    .= 1/cos(x+h) - 1/cos(x) by A1
    .= (1*cos(x)-1*cos(x+h))/(cos(x+h)*cos(x)) by A2,XCMPLX_1:130
    .= (cos(x)-cos(x+h))/((1/2)*(cos((x+h)+x)+cos((x+h)-x))) by SIN_COS4:32
    .= (cos(x)-cos(x+h))/(1/2)/(cos(2*x+h)+cos(h)) by XCMPLX_1:78
    .= 2*((cos(x)-cos(x+h))/(cos(2*x+h)+cos(h)));
  hence thesis;
end;
