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,x3 are_mutually_distinct
  & x0>0 & x1>0 & x2>0 & x3>0 implies
  [!f,x0,x1,x2,x3!] = (sqrt x0+sqrt x1+sqrt x2+sqrt x3)/((sqrt x0+sqrt x1)
  *(sqrt x0+sqrt x2)*(sqrt x0+sqrt x3)*(sqrt x1+sqrt x2)*(sqrt x1+sqrt x3)
  *(sqrt x2+sqrt x3))
proof
  assume that
A1:for x holds f.x = sqrt x and
A2:x0,x1,x2,x3 are_mutually_distinct and
A3:x0>0 & x1>0 & x2>0 & x3>0;
A4:f.x0 = sqrt x0 & f.x1 = sqrt x1 & f.x2 = sqrt x2 & f.x3 = sqrt x3 by A1;
sqrt x0>0 & sqrt x1>0 & sqrt x2>0 & sqrt x3>0 by A3,SQUARE_1:25;
then A5:sqrt x0+sqrt x1>0 & sqrt x1+sqrt x2>0 & sqrt x2+sqrt x3>0
   & sqrt x2+sqrt x0>0 & sqrt x3+sqrt x1>0;
A6:x0<>x1 & x0<>x2 & x0<>x3 & x1<>x2 & x1<>x3 & x2<>x3 by A2,ZFMISC_1:def 6;
  then [!f,x0,x1,x2,x3!] = ((1/(sqrt x0 + sqrt x1)
       - (sqrt x1-sqrt x2)/(x1-x2))/(x0-x2)
       - ((sqrt x1-sqrt x2)/(x1-x2) - (sqrt x2-sqrt x3)/(x2-x3))/(x1-x3))
       /(x0-x3) by A3,A4,SQUARE_1:36
    .= ((1/(sqrt x0 + sqrt x1) - 1/(sqrt x1 + sqrt x2))/(x0-x2)
       - ((sqrt x1-sqrt x2)/(x1-x2) - (sqrt x2-sqrt x3)/(x2-x3))/(x1-x3))
       /(x0-x3) by A3,A6,SQUARE_1:36
    .= ((1/(sqrt x0 + sqrt x1) - 1/(sqrt x1 + sqrt x2))/(x0-x2)
       - (1/(sqrt x1 + sqrt x2) - (sqrt x2 - sqrt x3)/(x2-x3))/(x1-x3))/(x0-x3)
                                                     by A3,A6,SQUARE_1:36
    .= ((1/(sqrt x0 + sqrt x1) - 1/(sqrt x1 + sqrt x2))/(x0-x2)
       - (1/(sqrt x1 + sqrt x2) - 1/(sqrt x2 + sqrt x3))/(x1-x3))/(x0-x3)
                                                     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)
       - (1/(sqrt x1 + sqrt x2) - 1/(sqrt x2 + sqrt x3))/(x1-x3))/(x0-x3)
                                                           by A5,XCMPLX_1:130
    .= (((1*(sqrt x1 + sqrt x2)-1*(sqrt x0 + sqrt x1))
       /((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2)))/(x0-x2)
       - ((1*(sqrt x2 + sqrt x3)-1*(sqrt x1 + sqrt x2))
       /((sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x3)))/(x1-x3))/(x0-x3)
                                                          by A5,XCMPLX_1:130
    .= (((sqrt x2-sqrt x0)/((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2)))
       /(-(x2-x0))- ((sqrt x3-sqrt x1)/((sqrt x1+sqrt x2)*(sqrt x2+sqrt x3)))
       /(-(x3-x1)))/(x0-x3)
    .= (-((sqrt x2-sqrt x0)/((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2)))/(x2-x0)
       - ((sqrt x3-sqrt x1)/((sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x3)))
       /(-(x3-x1)))/(x0-x3) by XCMPLX_1:188
    .= (-(1/((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2)))*((sqrt x2-sqrt x0)
       /(x2-x0))- -((sqrt x3-sqrt x1)/((sqrt x1 + sqrt x2)
       *(sqrt x2 + sqrt x3)))/(x3-x1))/(x0-x3) by XCMPLX_1:188
    .= (-(1/((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2)))*(1/(sqrt x2+sqrt x0))
       - -((sqrt x3-sqrt x1)/((sqrt x1 + sqrt x2)
       *(sqrt x2 + sqrt x3)))/(x3-x1))/(x0-x3) by A3,A6,SQUARE_1:36
    .= (-1/((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0))
       + (1/((sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x3)))
       *((sqrt x3-sqrt x1)/(x3-x1)))/(x0-x3) by XCMPLX_1:102
    .= (-1/((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0))
       + (1/((sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x3)))
       *(1/(sqrt x3+sqrt x1)))/(x0-x3) by A3,A6,SQUARE_1:36
    .= (1/((sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1))
       -1/((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0)))
       /(x0-x3) by XCMPLX_1:102
    .= ((1*((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0))
       -1*((sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1)))
       /(((sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1))
       *((sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0))))
       /(x0-x3) by A5,XCMPLX_1:130
    .= (((sqrt x0 + sqrt x1)*(sqrt x2 + sqrt x0)
       -(sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1))*(sqrt x1 + sqrt x2))
       /((sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1)*(sqrt x0 + sqrt x1)
       *(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0)*(sqrt x1 + sqrt x2))/(x0-x3)
    .= (sqrt x0*sqrt x2+sqrt x0*sqrt x0+sqrt x1*sqrt x2+sqrt x1*sqrt x0
       -(sqrt x2*sqrt x3+sqrt x2*sqrt x1+sqrt x3*sqrt x3+sqrt x3*sqrt x1))
       /((sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1)*(sqrt x0 + sqrt x1)
       *(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0))/(x0-x3) by A5,XCMPLX_1:91
    .= (sqrt x0*sqrt x2+sqrt x0^2+sqrt x1*sqrt x2+sqrt x1*sqrt x0
       -(sqrt x2*sqrt x3+sqrt x2*sqrt x1+sqrt x3*sqrt x3+sqrt x3*sqrt x1))
       /((sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1)*(sqrt x0 + sqrt x1)
       *(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0))/(x0-x3) by A3,SQUARE_1:29
    .= ((sqrt x0*sqrt x2+x0+sqrt x1*sqrt x2+sqrt x1*sqrt x0)
       -(sqrt x2*sqrt x3+sqrt x2*sqrt x1+(sqrt x3*sqrt x3)+sqrt x3*sqrt x1))
       /((sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1)*(sqrt x0 + sqrt x1)
       *(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0))/(x0-x3) by A3,SQUARE_1:22
    .= ((sqrt x0*sqrt x2+x0+sqrt x1*sqrt x2+sqrt x1*sqrt x0)
       -(sqrt x2*sqrt x3+sqrt x2*sqrt x1+sqrt x3^2+sqrt x3*sqrt x1))
       /((sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1)*(sqrt x0 + sqrt x1)
       *(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0))/(x0-x3) by A3,SQUARE_1:29
    .= (sqrt x0*sqrt x2+x0+sqrt x1*sqrt x2+sqrt x1*sqrt x0
       -(sqrt x2*sqrt x3+sqrt x2*sqrt x1+x3+sqrt x3*sqrt x1))
       /((sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1)*(sqrt x0 + sqrt x1)
       *(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0))/(x0-x3) by A3,SQUARE_1:22
    .= ((sqrt x2+sqrt x1)*(sqrt x0-sqrt x3)+(x0-x3))
       /((sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1)*(sqrt x0 + sqrt x1)
       *(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0)*(x0-x3)) by XCMPLX_1:78
    .= ((sqrt x0-sqrt x3)*(sqrt x2+sqrt x1)
       +(sqrt x0-sqrt x3)*(sqrt x0+sqrt x3))
       /((sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1)*(sqrt x0 + sqrt x1)
       *(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0)*(x0-x3)) by A3,SQUARE_1:35
    .= ((sqrt x0-sqrt x3)*(sqrt x2+sqrt x1+sqrt x0+sqrt x3))
       /((sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1)*(sqrt x0 + sqrt x1)
       *(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0)*(x0-x3))
    .= ((sqrt x0-sqrt x3)/(x0-x3))*((sqrt x2+sqrt x1+sqrt x0+sqrt x3)
       /((sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1)*(sqrt x0 + sqrt x1)
       *(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0))) by XCMPLX_1:76
    .= (1/(sqrt x0+sqrt x3))*((sqrt x2+sqrt x1+sqrt x0+sqrt x3)
       /((sqrt x2 + sqrt x3)*(sqrt x3 + sqrt x1)*(sqrt x0 + sqrt x1)
       *(sqrt x1 + sqrt x2)*(sqrt x2 + sqrt x0))) by A3,A6,SQUARE_1:36
    .= (sqrt x2+sqrt x1+sqrt x0+sqrt x3)/((sqrt x2 + sqrt x3)
       *(sqrt x3 + sqrt x1)*(sqrt x0 + sqrt x1)*(sqrt x1 + sqrt x2)
       *(sqrt x2 + sqrt x0)*(sqrt x0+sqrt x3)) by XCMPLX_1:103
    .= (sqrt x0+sqrt x1+sqrt x2+sqrt x3)/((sqrt x0+sqrt x1)*(sqrt x0+sqrt x2)
       *(sqrt x0+sqrt x3)*(sqrt x1+sqrt x2)*(sqrt x1+sqrt x3)
       *(sqrt x2+sqrt x3));
  hence thesis;
end;
