reserve x,y,a,b,c,p,q for Real;
reserve m,n for Element of NAT;

theorem
  a<>0 & b^2+2*a*b+5*a^2-4*a*c>0 & Polynom(a,b,c,c,b,a,x)=0 implies for
  y1,y2 being Real st y1 = (a-b+sqrt(b^2+2*a*b+5*a^2-4*a*c))/(2*a) & y2 = (a-b-
sqrt(b^2+2*a*b+5*a^2-4*a*c))/(2*a) holds x=-1 or x = (y1 + sqrt delta(1,(-y1),1
))/2 or x = (y2 + sqrt delta(1,(-y2),1))/2 or x = (y1 - sqrt delta(1,(-y1),1))/
  2 or x = (y2 - sqrt delta(1,(-y2),1))/2
proof
  assume that
A1: a<>0 & b^2+2*a*b+5*a^2-4*a*c>0 and
A2: Polynom(a,b,c,c,b,a,x)=0;
  let y1,y2 be Real;
  assume that
A3: y1 = (a-b+sqrt(b^2+2*a*b+5*a^2-4*a*c))/(2*a) and
A4: y2 = (a-b-sqrt(b^2+2*a*b+5*a^2-4*a*c))/(2*a);
A5: 0=(x|^5+1)*a+(x|^4+x+0)*b+(c*(x|^3)+c*(x^2)+0*c) by A2
    .=(x|^5+1|^5)*a+(x|^(3+1)+x)*b+(x|^3+x^2)*c
    .=(x|^5+1|^5)*a+(x|^3*x+x)*b+(x|^(2+1)+x^2)*c by NEWTON:6
    .=(x|^5+1|^5)*a+(x|^3+1+0)*x*b+(x*x|^(1+1)+1*x^2+0*x^2)*c by NEWTON:6
    .=(x|^5+1|^5)*a+(x|^3+1+0)*x*b+(x*(x|^1*x)+1*x^2+0*x^2)*c by NEWTON:6
    .=(x|^5+1|^5)*a+(x|^3+1+0)*x*b+(x*x^2+1*x^2+0*x^2)*c
    .=((x+1)*(x|^4-x|^3*1+x|^2*1|^2-x*1|^3+1|^4))*a +(x|^3+1)*x*b+(x+1+0)*x
  ^2*c by Th11
    .=((x+1)*(x|^4-x|^3+x|^2*1-x*1|^3+1|^4))*a +(x|^3+1)*x*b+(x+1+0)*x^2*c
    .=((x+1)*(x|^4-x|^3+x|^2-x*1+1|^4))*a+(x|^3+1)*x*b+(x+1+0)*x^2*c
    .=((x+1)*(x|^4-x|^3+x|^2-x+1))*a+(x|^3+1)*x*b+(x+1+0)*x^2*c
    .=((x+1)*(x|^4-x|^3+x|^2-x+1))*a+(x|^3+1|^3)*x*b+(x+1)*x^2*c
    .=((x+1)*(x|^4-x|^3+x|^2-x+1))*a+((x+1)*(x^2-x*1+1^2))*x*b +(x+1)*x^2*c
  by Th11
    .=(a*x|^4-a*x|^3+a*x|^2-a*x+a+(x*x*x*b-x*x*b+(b*x)) +(x*x*c))*(x+1)
    .=(a*x|^4-a*x|^3+a*x|^2-a*x+a+(x|^1*x*x*b-x*x*b+(b*x)) +(x*x*c))*(x+1)
    .=(a*x|^4-a*x|^3+a*x|^2-a*x+a+(x|^1*x*x*b-x*x*b+(b*x)) +(x|^1*x*c))*(x+1
  ) 
    .=(a*x|^4-a*x|^3+a*x|^2-a*x+a+(x|^1*x*x*b-x*x*b+(b*x)) +(x|^(1+1)*c))*(x
  +1) by NEWTON:6
    .=(a*x|^4-a*x|^3+a*x|^2-a*x+a+(x|^(1+1)*x*b-x*x*b+(b*x)) +(x|^2*c))*(x+1
  ) by NEWTON:6
    .=(a*x|^4-a*x|^3+a*x|^2-a*x+a+(x|^(2+1)*b-x*x*b+b*x) +(x|^2*c))*(x+1) by
NEWTON:6
    .=(a*x|^4-a*x|^3+a*x|^2-a*x+a+(x|^3*b-x|^1*x*b+b*x) +(x|^2*c))*(x+1) 
    .=(a*x|^4-a*x|^3+a*x|^2-a*x+a+(x|^3*b-x|^(1+1)*b+b*x) +(x|^2*c))*(x+1)
  by NEWTON:6
    .=(a*x|^4-(a-b)*x|^3+(a+c-b)*x|^2-(a-b)*x+a)*(x+1);
  now
    per cases by A5;
    case
      x+1=0;
      hence thesis;
    end;
    case
A6:   a*x|^4-(a-b)*x|^3+(a+c-b)*x|^2-(a-b)*x+a=0;
      set y=x+1/x;
      0=a*x|^4+(-a+b)*x|^3+(a+c-b)*x|^(1+1)+(-a+b)*x+a by A6
        .=a*x|^4+(-a+b)*x|^3+(a+c-b)*(x|^1*x)+(-a+b)*x+a by NEWTON:6
        .=a*x|^4+(-a+b)*x|^3+(a+c-b)*x^2+(-a+b)*x+a;
      then
A7:   Polynom(a,-a+b,a+c-b,-a+b,a,x)=0 by POLYEQ_2:def 1;
      y=x+1/x & y1 = (-(-a+b)+sqrt((-a+b)^2-4*a*(a+c-b)+8*a^2))/(2*a) by A3;
      hence thesis by A1,A4,A7,POLYEQ_2:3;
    end;
  end;
  hence thesis;
end;
