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