reserve a,b for Complex;
reserve z for Complex;
reserve n0 for non zero Nat;
reserve a0,a1,a2,s1,s2 for Complex;

theorem Th12:
  a2<>0 implies (a2*z|^2+a1*z+a0 = 0 iff z = -a1/(2*a2)+2-root(
  delta(a0,a1,a2))/(2*a2) or z = -a1/(2*a2)-2-root(delta(a0,a1,a2))/(2*a2))
proof
  set s1 = -a1/(2*a2)+2-root(delta(a0,a1,a2))/(2*a2);
  set s2 = -a1/(2*a2)-2-root(delta(a0,a1,a2))/(2*a2);
A1: -(s1+s2) = a1/(2*a2)+a1/(2*a2) .= 1/2*(a1/a2)+a1/(2*a2) by XCMPLX_1:103
    .= 1/2*(a1/a2)+1/2*(a1/a2) by XCMPLX_1:103
    .= a1/a2;
  assume
A2: a2<>0;
  then a2*z|^2+a1*z+a0 = (a2*z|^2+a1*z+a0)/a2*a2 by XCMPLX_1:87
    .= (a2*z|^2/a2+a1*z/a2+a0/a2)*a2 by XCMPLX_1:63
    .= (z|^2+a1*z/a2+a0/a2)*a2 by A2,XCMPLX_1:89
    .= (z|^2+a1*(z/a2)+a0/a2)*a2 by XCMPLX_1:74
    .= (z|^2+z/(a2/a1)+a0/a2)*a2 by XCMPLX_1:81
    .= (z|^2+(a1/a2)*z+a0/a2)*a2 by XCMPLX_1:79;
  then
A3: a2*z|^2+a1*z+a0 = 0 iff z|^2+(a1/a2)*z+a0/a2 = 0 by A2;
  s1*s2 = (a1/(2*a2))*(a1/(2*a2))- (2-root(delta(a0,a1,a2))/(2*a2))* (2
  -root(delta(a0,a1,a2))/(2*a2))
    .= (a1*a1)/((2*a2)*(2*a2)) - (2-root(delta(a0,a1,a2))/(2*a2))* (2-root(
  delta(a0,a1,a2))/(2*a2)) by XCMPLX_1:76
    .= (a1*a1)/(4*a2*a2) - ((2-root(delta(a0,a1,a2))* (2-root(delta(a0,a1,a2
  )))))/((2*a2)*(2*a2)) by XCMPLX_1:76
    .= (a1*a1)/(4*a2*a2) - (2-root(delta(a0,a1,a2))|^2)/ ((2*a2)*(2*a2)) by Th1
    .= (a1*a1)/(4*a2*a2) - delta(a0,a1,a2)/((2*a2)*(2*a2)) by Th7
    .= (a1*a1 - delta(a0,a1,a2))/(4*a2*a2) by XCMPLX_1:120
    .= (a0*(4*a2))/(a2*(4*a2))
    .= (a0/a2)*(4*a2/(4*a2)) by XCMPLX_1:76
    .= (a0/a2)*1 by A2,XCMPLX_1:60
    .= a0/a2;
  hence thesis by A3,A1,Th11;
end;
