
theorem Th2:
  for a,b,c,x,y being Real st a <> 0 & y = x + 1/x holds
  Polynom(a,b,c,b,a,x) = 0 implies x <> 0 & a*y^2 + b*y + c - 2*a = 0
proof
  let a,b,c,x,y be Real;
  assume that
A1: a <> 0 and
A2: y = x + 1/x;
  assume
A3: Polynom(a,b,c,b,a,x) = 0;
A4: x <> 0
  proof
    assume x = 0;
    then a*(0 to_power 4)+b*(0|^ 3)+ a = 0 by A3;
    then a*0+b*(0|^ 3)+ a = 0 by POWER:def 2;
    then a*0+b*0+ a = 0 by NEWTON:11;
    hence contradiction by A1;
  end;
  then
A5: x^2 > 0 by SQUARE_1:12;
A6: x|^ 4 = x to_power (2+2) by POWER:41;
A7: now
    per cases by A4,XXREAL_0:1;
    case
A8:   x > 0;
      set n = -(b*x+a);
      set m = (a*x^2)+(b*x+c);
      x|^ 3 = (x to_power(2+1)) by POWER:41
        .= (x to_power 2)*(x to_power 1) by A8,POWER:27;
      then
A9:   x|^ 3 = (x to_power 2)*x
        .= x^2*x by POWER:46;
      x|^ 4 = (x to_power 2)*(x to_power 2) by A6,A8,POWER:27
        .= x^2*(x to_power 2) by POWER:46
        .= x^2*x^2 by POWER:46;
      then m*x^2 = n*1 by A3,A9;
      then m/1 = n/x^2 by A5,XCMPLX_1:94;
      then (a*x^2)+(b*x+c) = (-(b*x+a))*(1/x^2) by XCMPLX_1:99
        .= (-(b*x+a))*(x^2)" by XCMPLX_1:215
        .= -b*(x*(x^2)")-a*(x^2)";
      then a*(x^2+(x^2)") = -(b*(x*(x^2)"+x))- c;
      then
A10:  a*(x^2+1/x^2) = -(b*(x*(x^2)"+x))- c by XCMPLX_1:215
        .= -(b*(x*(1/x^2)+x))- c by XCMPLX_1:215;
      1/(x*x) = (1/x)*(1/x) by XCMPLX_1:102;
      then a*(x^2+1/x^2) = -(b*((x*(1/x))*(1/x)+x))- c by A10;
      then
A11:  a*(x^2+1/x^2) = -(b*(1 *(1/x)+x))- c by A8,XCMPLX_1:106;
      x*y = x*x +x*(1/x) by A2;
      then x*y + 0 = (x^2 + 1) by A4,XCMPLX_1:106;
      hence a*(x^2+1/x^2) = -(b*(x+1/x))- c & x^2 - x*y + 1 = 0 by A11;
    end;
    case
A12:  x < 0;
      set n = -(b*x+a);
      set m = (a*x^2)+(b*x+c);
      (-x)|^ 3 +(x|^ 3) = -((x|^ 3)+-(x|^ 3)) by Lm2,POWER:2
        .= (x|^ 3)-(x|^ 3);
      then
A13:  x|^ 3 = -((-x)|^ 3);
A14:  0 < - x by A12,XREAL_1:58;
      (-x)|^ 4 = x|^ 4 by Lm1,POWER:1;
      then
A15:  x|^ 4 = (-x) to_power (2+2) by POWER:41
        .= ((-x) to_power 2)*((-x) to_power 2) by A14,POWER:27
        .= (-x)^2*((-x) to_power 2) by POWER:46
        .= x^2*(-x)^2 by POWER:46;
      (-x)|^ 3 = ((-x)to_power(2+1)) by POWER:41
        .= ((-x) to_power 2)*((-x) to_power 1) by A14,POWER:27;
      then
A16:  (-x)|^ 3 = ((-x) to_power 2)*(-x);
      (-x) to_power 2 = (-x)^2 by POWER:46
        .= x^2;
      then m*x^2 = n*1 by A3,A15,A16,A13;
      then m/1 = n/x^2 by A5,XCMPLX_1:94;
      then m = n*(1/x^2) by XCMPLX_1:99
        .= n*(x^2)" by XCMPLX_1:215
        .= -b*(x*(x^2)")-a*(x^2)";
      then a*(x^2+(x^2)") = -(b*(x*(x^2)"+x))- c;
      then a*(x^2+1/x^2) = -(b*(x*(x^2)"+x))- c by XCMPLX_1:215
        .= -(b*(x*(1/x^2)+x))- c by XCMPLX_1:215;
      then a*(x^2+1/x^2) = -(b*(x*((1/x)*(1/x))+x))- c by XCMPLX_1:102
        .= -(b*((x*(1/x))*(1/x)+x))- c;
      then
A17:  a*(x^2+1/x^2) = -(b*((1)*(1/x)+x))- c by A12,XCMPLX_1:106;
      x*y = x*x +x*(1/x) by A2
        .= x*x + 1 by A12,XCMPLX_1:106;
      hence a*(x^2+1/x^2) = -(b*(x+1/x))- c & x^2 - x*y + 1 = 0 by A17;
    end;
  end;
  y^2 = x^2+2*(x*(1/x))+(1/x)^2 by A2
    .= x^2+ 2*1 +(1/x)^2 by A4,XCMPLX_1:106
    .= x^2+ 2 +(1^2/x^2) by XCMPLX_1:76
    .= x^2-(- 2 -(1/x^2));
  then a*y^2 - 2*a = -(b*y)- c by A2,A7;
  hence thesis by A4;
end;
