reserve h,r,r1,r2,x0,x1,x2,x3,x4,x5,x,a,b,c,k for Real,
  f,f1,f2 for Function of REAL,REAL;

theorem Th27:
  (for x holds f.x = a*x^2+b*x+c) & x0<>x1 implies [!f,x0,x1!]=a*( x0+x1)+b
proof
  assume that
A1: for x holds f.x = a*x^2+b*x+c and
A2: x0<>x1;
A3: x0-x1<>0 by A2;
  [!f,x0,x1!] = (a*x0^2+b*x0+c-f.x1)/(x0-x1) by A1
    .= (a*x0^2+b*x0+c-(a*x1^2+b*x1+c))/(x0-x1) by A1
    .= (a*(x0+x1)+b)*(x0-x1)/(x0-x1);
  hence thesis by A3,XCMPLX_1:89;
end;
