reserve a,b,c,d,a9,b9,c9,d9,y,x1,u,v for Real,
  s,t,h,z,z1,z2,z3,s1,s2,s3 for Complex;

theorem Th1:
  a <> 0 & delta(a,b,c) >=0 & Polynom(a,b,c,z) = 0 implies z= (-b+
  sqrt delta(a,b,c))/(2*a) or z= (-b-sqrt delta(a,b,c))/(2*a) or z= -b/(2*a)
proof
A1: a = a+0*<i>;
  set y=Im z;
A2: z = Re z + (Im z)*<i> by COMPLEX1:13;
  set x=Re z;
  assume that
A3: a <> 0 and
A4: delta(a,b,c) >= 0;
  assume Polynom(a,b,c,z) = 0;
  then (a+0*<i>)*(x^2-y^2+2*(x*y)*<i>)+b*z+c = 0 by A2;
  then
  0 = Re a*Re (x^2-y^2+(2*x*y)*<i>) -Im a*Im (x^2-y^2+2*x*y*<i>)+ (Re a*Im
  (x^2-y^2+2*x*y*<i>) +Re (x^2-y^2+2*x*y*<i>)*Im a)*<i> +b*z+c by
COMPLEX1:82
    .=(a*Re (x^2-y^2+2*x*y*<i>)-Im a* Im (x^2-y^2+2*x*y*<i>))+(Re a*Im (x^2-
  y^2+2*x*y*<i>) +Re (x^2-y^2+2*x*y*<i>)*Im a)*<i> +b*z+c by A1,COMPLEX1:12
    .=(a*(x^2-y^2)-Im a* Im (x^2-y^2+2*x*y*<i>))+(Re a*Im (x^2-y^2+2*x*y*<i>
  ) +Re (x^2-y^2+2*x*y*<i>)*Im a)*<i> +b*z+c by COMPLEX1:12
    .=(a*(x^2-y^2)-0*Im (x^2-y^2+2*x*y*<i>) )+(Re a*Im (x^2-y^2+2*x*y*<i>)+
  Re (x^2-y^2+2*x*y*<i>) *Im a )*<i> +b*z+c by A1,COMPLEX1:12
    .=(a*(x^2-y^2)-0)+(Re a*(2*x*y)+Re (x^2-y^2+2*x*y*<i>) *Im a )*<i> +b*z+
  c by COMPLEX1:12
    .=(a*(x^2-y^2)-0)+(a*(2*x*y)+Re (x^2-y^2+2*x*y*<i>) *Im a )*<i> +b*z+c
  by A1,COMPLEX1:12
    .=(a*(x^2-y^2)-0)+(a*(2*x*y)+(x^2-y^2) *Im a )*<i> +b*z+c by COMPLEX1:12
    .=(a*(x^2-y^2)-0)+(a*(2*x*y)+(x^2-y^2)*0)*<i> +b*z+c by A1,COMPLEX1:12;
  then
A5: (a*(x^2-y^2)-0*(2*x*y))+(0+a*(2*x*y))*<i> +(b+0*<i>)*(x+y*<i>)+c = 0 by
COMPLEX1:13;
  then
A6: a*(x^2-y^2)+b*x+c+(a*(2*x*y)+b*y)*<i> = 0;
  then
A7: (2*a*x+b)*y = 0 by COMPLEX1:4,12;
  per cases by A7;
  suppose
A8: y = 0;
    then Polynom(a,b,c,x)=0 by A5;
    then
    Re z=(-b+sqrt delta(a,b,c))/(2*a) & Im z = 0 or Re z= (-b-sqrt delta(
    a,b,c))/(2*a) & Im z = 0 by A3,A4,A8,POLYEQ_1:5;
    then
    z=(-b+sqrt delta(a,b,c))/(2*a)+0*<i> or z= (-b-sqrt delta(a,b,c))/(2*
    a)+0*<i> by COMPLEX1:13;
    hence thesis;
  end;
  suppose
    2*a*x+b = 0;
    then
A9: x= (-b)/(2*a) by A3,XCMPLX_1:89;
    then a*((b/(2*a))^2-y^2)+b*(-b/(2*a))+c = 0 by A6,COMPLEX1:4,12;
    then (b/(2*a))^2-y^2 = (-(b*(-b/(2*a))+c))/a-0 by A3,XCMPLX_1:89;
    then (b/(2*a))^2- (-(b*(-b/(2*a))+c))/a= y^2-0;
    then y^2 = (b/(2*a))^2+c*a"-((b^2/(2*a))*a");
    then y^2 *((2*a)^2)=(b^2/(2*a)^2+c*a"-((b^2/(2*a))*a"))*((2*a)^2) by
XCMPLX_1:76
      .=b^2/((2*a)^2)*((2*a)^2)+(c*a")*(2*a)^2 -(b^2*(2*a)"*a")*(2*a)^2;
    then
A10: y
^2 *((2*a)^2)=b^2+(c*a")*(2*a)^2-(b^2*((2*a)"*a"))*(2*a)^2 by A3,XCMPLX_1:87
      .=b^2+(c*a")*(2*a)^2-(b^2*((2*a)*a)")*(2*a)^2 by XCMPLX_1:204
      .=b^2+(c*a")*(2*a)^2-(b^2*(2*(a*a))")*(2*a)^2;
    set t = (b^2*(2*(a*a))")*(2*a)^2;
    t*((2*a)^2)" = b^2*(2*(a*a))"*(((2*a)^2)*(1/(2*a)^2));
    then t*((2*a)^2)" = b^2*(2*(a*a))"*1 by A3,XCMPLX_1:106;
    then t*((2*a)^2)"*2" = b^2*(2*a^2)"*2";
    then t*((2*a)^2)"*2" = b^2*((2*a^2)"*2");
    then t*((2*a)^2)"*2" = b^2*((2*(a^2*2))") by XCMPLX_1:204;
    then (t*2")/(2*a)^2*(2*a)^2 = b^2/(2*a)^2*(2*a)^2;
    then t*2" = b^2/(2*a)^2*(2*a)^2 by A3,XCMPLX_1:87;
    then
A11: t/2 = b^2 by A3,XCMPLX_1:87;
    set t=(c*a")*(2*a)^2;
    t=(c/a*a)*2*(2*a);
    then
A12: t=c*2*(2*a) by A3,XCMPLX_1:87;
    -delta(a,b,c) <= 0 by A4;
    then (y *(2*a))^2 = 0 by A10,A11,A12,XREAL_1:63;
    then Im z=0 by A3;
    then z = -b/(2*a)+0*<i> by A9,COMPLEX1:13;
    hence thesis;
  end;
end;
