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<>0 & x1<>0 & x2<>0
  & x0,x1,x2 are_mutually_distinct implies
  [!f,x0,x1,x2!] = (k/(x0*x1*x2))*(1/x0+1/x1+1/x2)
proof
  assume that
A1:for x holds f.x = k/(x^2) and
A2:x0<>0 & x1<>0 & x2<>0 and
A3:x0,x1,x2 are_mutually_distinct;
A4:f.x0 = k/(x0^2) & f.x1 = k/(x1^2) & f.x2 = k/(x2^2) by A1;
A5:x1-x0<>0 & x2-x1<>0 & x0-x2<>0 by A3,ZFMISC_1:def 5;
  [!f,x0,x1,x2!] = ((k*(1/(x0^2)-1/(x1^2)))/(x0-x1)
       -(k*(1/(x1^2)-1/(x2^2)))/(x1-x2))/(x0-x2) by A4
    .= ((k*((1*x1^2-1*x0^2)/((x0^2)*(x1^2))))/(x0-x1)
       -(k*(1/(x1^2)-1/(x2^2)))/(x1-x2))/(x0-x2) by A2,XCMPLX_1:130
    .= ((k*((1*x1^2-1*x0^2)/((x0^2)*(x1^2))))/(x0-x1)
       -(k*((1*x2^2-1*x1^2)/((x1^2)*(x2^2))))/(x1-x2))/(x0-x2)
                                                   by A2,XCMPLX_1:130
    .= (k*((((x1-x0)*(x1+x0))/((x0^2)*(x1^2)))/(-(x1-x0)))
       -k*((((x2-x1)*(x2+x1))/((x1^2)*(x2^2)))/(-(x2-x1))))/(x0-x2)
    .= (k*(-(((x1-x0)*(x1+x0))/((x0^2)*(x1^2)))/(x1-x0))
       -k*((((x2-x1)*(x2+x1))/((x1^2)*(x2^2)))/(-(x2-x1))))/(x0-x2)
                                                    by XCMPLX_1:188
    .= (-k*((((x1-x0)*(x1+x0))/((x0^2)*(x1^2)))/(x1-x0))
       -k*(-(((x2-x1)*(x2+x1))/((x1^2)*(x2^2)))/(x2-x1)))/(x0-x2)
                                                    by XCMPLX_1:188
    .= (-k*(((x1-x0)*(x1+x0))/(x1-x0)/((x0^2)*(x1^2)))
       +k*(((x2-x1)*(x2+x1))/(x2-x1)/((x1^2)*(x2^2))))/(x0-x2)
    .= (-k*((x1+x0)/((x0^2)*(x1^2)))
       +k*(((x2-x1)*(x2+x1))/(x2-x1)/((x1^2)*(x2^2))))/(x0-x2)
                                                       by A5,XCMPLX_1:89
    .= (-k*(x1/(x0*x0*x1*x1)+x0/(x0*x0*x1*x1))
       +k*((x2+x1)/(x1*x1*x2*x2)))/(x0-x2) by A5,XCMPLX_1:89
    .= (-k*(1/(x0*x0*x1)*(x1/x1)+x0/(x0*(x0*x1*x1)))
       +k*(x2/(x2*(x1*x1*x2))+x1/(x1*(x1*x2*x2))))/(x0-x2) by XCMPLX_1:103
    .= (-k*(1/(x0*x0*x1)*(x1/x1)+x0/(x0*(x0*x1*x1)))
       +k*(1/(x1*x1*x2)*(x2/x2)+x1/(x1*(x1*x2*x2))))/(x0-x2) by XCMPLX_1:103
    .= (-k*(1/(x0*x0*x1)*(x1/x1)+1/(x0*x1*x1)*(x0/x0))
       +k*(1/(x1*x1*x2)*(x2/x2)+x1/(x1*(x1*x2*x2))))/(x0-x2) by XCMPLX_1:103
    .= (-k*(1/(x0*x0*x1)*(x1/x1)+1/(x0*x1*x1)*(x0/x0))
       +k*(1/(x1*x1*x2)*(x2/x2)+1/(x1*x2*x2)*(x1/x1)))/(x0-x2) by XCMPLX_1:103
    .= (-k*(1/(x0*x0*x1)*1+1/(x0*x1*x1)*(x0/x0))
       +k*(1/(x1*x1*x2)*(x2/x2)+1/(x1*x2*x2)*(x1/x1)))/(x0-x2)
                                                            by A2,XCMPLX_1:60
    .= (-k*(1/(x0*x0*x1)*1+1/(x0*x1*x1)*(x0/x0))
       +k*(1/(x1*x1*x2)*1+1/(x1*x2*x2)*(x1/x1)))/(x0-x2) by A2,XCMPLX_1:60
    .= (-k*(1/(x0*x0*x1)*1+1/(x0*x1*x1)*1)
       +k*(1/(x1*x1*x2)*1+1/(x1*x2*x2)*(x1/x1)))/(x0-x2) by A2,XCMPLX_1:60
    .= (-k*(1/(x0*(x0*x1))+1/((x0*x1)*x1))
       +k*(1/(x1*(x1*x2))+1/((x1*x2)*x2)))/(x0-x2) by A2,XCMPLX_1:60
    .= (-k*((1/x0)*(1/(x0*x1))+1/((x0*x1)*x1))
       +k*(1/(x1*(x1*x2))+1/((x1*x2)*x2)))/(x0-x2) by XCMPLX_1:102
    .= (-k*((1/x0)*(1/(x0*x1))+1/((x0*x1)*x1))
       +k*((1/x1)*(1/(x1*x2))+1/((x1*x2)*x2)))/(x0-x2) by XCMPLX_1:102
    .= (-k*((1/x0)*(1/(x0*x1))+(1/(x0*x1))*(1/x1))
       +k*((1/x1)*(1/(x1*x2))+1/((x1*x2)*x2)))/(x0-x2) by XCMPLX_1:102
    .= (-k*(1/(x0*x1))*(1/x0+1/x1)
       +k*((1/x1)*(1/(x1*x2))+(1/(x1*x2))*(1/x2)))/(x0-x2) by XCMPLX_1:102
    .= k*(1/(x1*x2)/(x0-x2)*(1/x1+1/x2)
       -1/(x0*x1)/(x0-x2)*(1/x0+1/x1))
    .= k*(1/(x1*x2*(x0-x2))*(1/x1+1/x2)
       -1/(x0*x1)/(x0-x2)*(1/x0+1/x1)) by XCMPLX_1:78
    .= k*(1/(x1*x2*(x0-x2))*(1/x1+1/x2)
       -1/(x0*x1*(x0-x2))*(1/x0+1/x1)) by XCMPLX_1:78
    .= k*(1/(x1*x2*(x0-x2))*((1*x2+1*x1)/(x1*x2))
       -1/(x0*x1*(x0-x2))*(1/x0+1/x1)) by A2,XCMPLX_1:116
    .= k*(1/(x1*x2*(x0-x2))*((1*x2+1*x1)/(x1*x2))
       -1/(x0*x1*(x0-x2))*((1*x1+1*x0)/(x0*x1))) by A2,XCMPLX_1:116
    .= k*(1/(x1*x2*(x0-x2))/((x1*x2)/(x2+x1))
       -1/(x0*x1*(x0-x2))*((x1+x0)/(x0*x1))) by XCMPLX_1:79
    .= k*(1/(x1*x2*(x0-x2))/((x1*x2)/(x2+x1))
       -1/(x0*x1*(x0-x2))/((x0*x1)/(x1+x0))) by XCMPLX_1:79
    .= k*(1/(x1*x2*(x0-x2))/(x1*x2)*(x2+x1)
       -1/(x0*x1*(x0-x2))/((x0*x1)/(x1+x0))) by XCMPLX_1:82
    .= k*(1/(x1*x2*(x0-x2))/(x1*x2)*(x2+x1)
       -1/(x0*x1*(x0-x2))/(x0*x1)*(x1+x0)) by XCMPLX_1:82
    .= k*((x2+x1)/(x1*x2*(x0-x2))/(x1*x2)
       -(x1+x0)/(x0*x1*(x0-x2))/(x0*x1))
    .= k*((x2+x1)/((x1*x2*(x0-x2))*(x1*x2))
       -(x1+x0)/(x0*x1*(x0-x2))/(x0*x1)) by XCMPLX_1:78
    .= k*((x2+x1)/((x1*x2*(x0-x2))*(x1*x2))
       -(x1+x0)/((x0*x1*(x0-x2))*(x0*x1))) by XCMPLX_1:78
    .= k*((x2+x1)/((x1^2*(x0-x2))*x2^2)
       -(x1+x0)/((x1^2*(x0-x2))*x0^2))
    .= k*((x2+x1)/(x1^2*(x0-x2))/x2^2
       -(x1+x0)/((x1^2*(x0-x2))*x0^2)) by XCMPLX_1:78
    .= k*(1/(x1^2*(x0-x2))*((x2+x1)/x2^2)
       -(x1+x0)/(x1^2*(x0-x2))/x0^2) by XCMPLX_1:78
    .= k*(1/(x1^2*(x0-x2))*(((x2+x1)/x2^2)-((x1+x0)/x0^2)))
    .= k*(1/(x1^2*(x0-x2))*((((x2+x1)*x0^2)-((x1+x0)*x2^2))
       /(x2^2*x0^2))) by A2,XCMPLX_1:130
    .= k*((1*(x0-x2))/(x1^2*(x0-x2))*(x1*(x0+x2)+x0*x2)
       /(x2^2*x0^2))
    .= k*(1/x1^2*(x1*x0+x1*x2+x0*x2)/(x2^2*x0^2)) by A5,XCMPLX_1:91
    .= k*((x1*x0+x1*x2+x0*x2)/x1^2/(x2^2*x0^2))
    .= k*((x1*x0+x1*x2+x0*x2)/(x1^2*(x2^2*x0^2))) by XCMPLX_1:78
    .= k*((1*(x1*x0))/((x1*x2^2*x0)*(x1*x0))+(1*(x1*x2))/((x1*x2*x0^2)*(x1*x2))
       +(1*(x0*x2))/((x1^2*x2*x0)*(x0*x2)))
    .= k*(1/(x1*x2^2*x0)+(1*(x1*x2))/((x1*x2*x0^2)*(x1*x2))
       +(1*(x0*x2))/((x1^2*x2*x0)*(x0*x2))) by A2,XCMPLX_1:91
    .= k*(1/(x1*x2^2*x0)+1/(x1*x2*x0^2)
       +(1*(x0*x2))/((x1^2*x2*x0)*(x0*x2))) by A2,XCMPLX_1:91
    .= k*((1/((x1*x2*x0)*x2))+(1/((x1*x2*x0)*x0))
       +(1/((x1*x2*x0)*x1))) by A2,XCMPLX_1:91
    .= k*((1/(x1*x2*x0))*(1/x2)+(1/((x1*x2*x0)*x0))
       +(1/((x1*x2*x0)*x1))) by XCMPLX_1:102
    .= k*((1/(x1*x2*x0))*(1/x2)+(1/(x1*x2*x0))*(1/x0)
       +(1/((x1*x2*x0)*x1))) by XCMPLX_1:102
    .= k*((1/(x1*x2*x0))*(1/x2)+(1/(x1*x2*x0))*(1/x0)
       +(1/(x1*x2*x0))*(1/x1)) by XCMPLX_1:102
    .= (k/(x0*x1*x2))*(1/x0+1/x1+1/x2);
  hence thesis;
end;
