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 bD(f,h).x = 2*(cos(x-h)-cos(x))/(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;
  bD(f,h).x = f.x - f.(x-h) by DIFF_1:4
    .= 1/cos(x) - f.(x-h) by A1
    .= 1/cos(x) - 1/cos(x-h) by A1
    .= (1*cos(x-h)-1*cos(x))/(cos(x)*cos(x-h)) by A2,XCMPLX_1:130
    .= (cos(x-h)-cos(x))/((1/2)*(cos(x+(x-h))+cos(x-(x-h)))) by SIN_COS4:32
    .= (cos(x-h)-cos(x))/(1/2)/(cos(2*x-h)+cos(h)) by XCMPLX_1:78
    .= 2*((cos(x-h)-cos(x))/(cos(2*x-h)+cos(h)));
  hence thesis;
end;
