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 = k/(x^2)) & x0<>x1 & x0<>0 & x1<>0
  implies [!f,x0,x1!] = -(k/(x0*x1))*(1/x0+1/x1)
proof
  assume that
A1: for x holds f.x = k/(x^2) and
A2: x0<>x1 & x0<>0 & x1<>0;
A3: x1-x0<>0 by A2;
f.x0 = k/(x0^2) & f.x1 = k/(x1^2) by A1;
  then [!f,x0,x1!] = (k*(1/(x0^2)-1/(x1^2)))/(x0-x1)
    .= (k*((1*x1^2-1*x0^2)/((x0^2)*(x1^2))))/(x0-x1) by A2,XCMPLX_1:130
    .= k*((((x1-x0)*(x1+x0))/((x0^2)*(x1^2)))/(-(x1-x0)))
    .= k*(-(((x1-x0)*(x1+x0))/((x0^2)*(x1^2)))/(x1-x0)) by XCMPLX_1:188
    .= -k*(((x1-x0)*(x1+x0))/(x1-x0)/((x0^2)*(x1^2)))
    .= -k*((x1+x0)/(x0*x0*x1*x1)) by A3,XCMPLX_1:89
    .= -k*(x1/(x1*(x0*x0*x1))+x0/(x0*(x0*x1*x1)))
    .= -k*(1/(x0*x0*x1)*(x1/x1)+x0/(x0*(x0*x1*x1))) by XCMPLX_1:103
    .= -k*(1/(x0*x0*x1)*(x1/x1)+1/(x0*x1*x1)*(x0/x0)) by XCMPLX_1:103
    .= -k*(1/(x0*x0*x1)*1+1/(x0*x1*x1)*(x0/x0)) by A2,XCMPLX_1:60
    .= -k*(1/(x0*(x0*x1))+1/((x0*x1)*x1)) by A2,XCMPLX_1:60
    .= -k*((1/x0)*(1/(x0*x1))+1/((x0*x1)*x1)) by XCMPLX_1:102
    .= -k*((1/x0)*(1/(x0*x1))+(1/(x0*x1))*(1/x1)) by XCMPLX_1:102
    .= -(k/(x0*x1))*(1/x0+1/x1);
  hence thesis;
end;
