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 Th2:
  a <> 0 & delta(a,b,c) < 0 & Polynom(a,b,c,z) = 0 implies z= -b/(2
*a)+(sqrt (-delta(a,b,c))/(2*a))*<i> or z= -b/(2*a)+(-sqrt (-delta(a,b,c))/(2*a
  ))*<i>
proof
  assume that
A1: a <> 0 and
A2: delta(a,b,c) < 0;
A3: a= a+0*<i>;
  now
    set t2=(-b^2+c*a*4)/4;
    let z;
    set x=Re z;
    set y=Im z;
A4: z = x + y*<i> by COMPLEX1:13;
    assume Polynom(a,b,c,z) = 0;
    then (a+0*<i>)*(x^2-y^2+(2*x*y)*<i>)+b*z+c = 0 by A4;
    then
    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 = 0 by
COMPLEX1:82;
    then
    (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 = 0 by A3,COMPLEX1:12;
    then (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 = 0 by COMPLEX1:12;
    then
    (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 = 0 by A3,COMPLEX1:12;
    then
    (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
    = 0 by COMPLEX1:12;
    then (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 =
    0 by A3,COMPLEX1:12;
    then (a*(x^2-y^2)-0)+(a*(2*x*y)+(x^2-y^2) *Im a)*<i>+b*z+c = 0 by
COMPLEX1:12;
    then (a*(x^2-y^2)-0)+(a*(2*x*y)+(x^2-y^2) *0)*<i>+b*z+c = 0 by A3,
COMPLEX1:12;
    then (a*(x^2-y^2)-0*(2*x*y))+((x^2-y^2)*0+a*(2*x*y))*<i> +(b+0*<i>)*(x+y*
    <i>)+c = 0 by COMPLEX1:13;
    then
A5: a*(x^2-y^2)+b*x+c+(a*(2*x*y)+b*y)*<i> = 0;
    then (a*(2*x))*y+b*y = 0 by COMPLEX1:4,12;
    then
A6: (a*2*x)*y = (-b)*y;
    set t = (b^2*(2*(a*a))")*(2*a)^2;
    set t1= (x*a+b/2)^2;
    0-delta(a,b,c)>0 by A2;
    then
A7: 0+0<t1+t2 by XREAL_1:8,63;
    -a*y^2+((b*x+a*x^2)+c )+a*y^2= 0+a*y^2 by A5,COMPLEX1:4,12;
    then a*x^2*a+b*x*a+c*a = a*y^2*a by XCMPLX_1:9;
    then y<>0 by A7;
    then a*(2*x) = -b by A6,XCMPLX_1:5;
    then 2*x = (-b)/a by A1,XCMPLX_1:89;
    then x = 1/a*((-b)/2);
    then
A8: x = (-b)/(2*a) by XCMPLX_1:103;
    then a*((b/(2*a))^2-y^2)+b*(-b/(2*a))+c = 0 by A5,COMPLEX1:4,12;
    then (b/(2*a))^2-y^2 = (-(b*(-b/(2*a))+c))/a-0 by A1,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
A9: y
^2 *((2*a)^2)=b^2+(c*a")*(2*a)^2-(b^2*((2*a)"*a"))*(2*a)^2 by A1,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;
    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 A1,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 A1,XCMPLX_1:87;
    then
A10: t/2 = b^2 by A1,XCMPLX_1:87;
    set t=(c*a")*(2*a)^2;
    t=(c/a*a)*2*(2*a);
    then t=c*2*(2*a) by A1,XCMPLX_1:87;
    then (y *(2*a))^2 =(sqrt (-delta(a,b,c)))^2 by A2,A9,A10,SQUARE_1:def 2;
    then
    ((y *(2*a))+(sqrt (-delta(a,b,c))))* ((y *(2*a))-(sqrt (-delta(a,b,c)
    )))=0;
    then
    ((y *(2*a))+(sqrt (-delta(a,b,c))))=0 or ((y *(2*a))-(sqrt (-delta(a,
    b,c))))=0;
    then
    y =(-sqrt (-delta(a,b,c)))/(2*a) or y *(2*a)/(2*a)=sqrt (-delta(a,b,c
    ))/(2*a) by A1,XCMPLX_1:89;
    then Re z=-b/(2*a) & Im z=sqrt (-delta(a,b,c))/(2*a) or Re z= -b/(2*a) &
    Im z= -sqrt (-delta(a,b,c))/(2*a) by A1,A8,XCMPLX_1:89;
    hence z= -b/(2*a)+(sqrt(-delta(a,b,c))/(2*a))*<i> or z= -b/(2*a)+(-sqrt (-
    delta(a,b,c))/(2*a))*<i> by COMPLEX1:13;
  end;
  hence thesis;
end;
