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 = sqrt x) & x0,x1,x2 are_mutually_distinct
  & x0>0 & x1>0 & x2>0 implies [!f,x0,x1,x2!]
  = -1/((sqrt x0+sqrt x1)*(sqrt x0+sqrt x2)*(sqrt x1+sqrt x2))
proof
  assume that
A1:for x holds f.x = sqrt x and
A2:x0,x1,x2 are_mutually_distinct and
A3:x0>0 & x1>0 & x2>0;
A4:f.x0 = sqrt x0 & f.x1 = sqrt x1 & f.x2 = sqrt x2 by A1;
sqrt x0>0 & sqrt x1>0 & sqrt x2>0 by A3,SQUARE_1:25;
then A5:sqrt x0+sqrt x1>0 & sqrt x1+sqrt x2>0;
A6:x0<>x1 & x1<>x2 & x2<>x0 by A2,ZFMISC_1:def 5;
  then
[!f,x0,x1,x2!] = (1/(sqrt x0 + sqrt x1) - (sqrt x1-sqrt x2)/(x1-x2))/(x0-x2)
                                                  by A3,A4,SQUARE_1:36
    .= (1/(sqrt x0 + sqrt x1) - 1/(sqrt x1 + sqrt x2))/(x0-x2)
                                                     by A3,A6,SQUARE_1:36
    .= ((1*(sqrt x1 + sqrt x2)-1*(sqrt x0 + sqrt x1))
       /((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2)))/(x0-x2) by A5,XCMPLX_1:130
    .= ((sqrt x2-sqrt x0)/((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2)))
       /(-(x2-x0))
    .= -((sqrt x2-sqrt x0)/((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2)))
       /(x2-x0) by XCMPLX_1:188
    .= -1/((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2))
       *((sqrt x2 - sqrt x0)/(x2-x0))
    .= -1/((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2))
       *(1/(sqrt x2 + sqrt x0)) by A3,A6,SQUARE_1:36
    .= -1/((sqrt x0+sqrt x1)*(sqrt x1+sqrt x2)*(sqrt x2+sqrt x0))
                                                       by XCMPLX_1:102;
  hence thesis;
end;
