reserve a,b for Complex;
reserve z for Complex;
reserve n0 for non zero Nat;
reserve a0,a1,a2,s1,s2 for Complex;
reserve a3,x,q,r,s,s3 for Complex;
reserve a4,p,s4 for Complex;

theorem Th26:
  1_root_of_quartic(a0,a1,a2,a3)+2_root_of_quartic(a0,a1,a2,a3)+
  3_root_of_quartic(a0,a1,a2,a3)+4_root_of_quartic(a0,a1,a2,a3) = -a3
proof
  per cases;
  suppose
A1: 8*a1 -4*a2*a3 + a3|^3 = 0;
    then
A2: ( ex p,r,s1 st p = (8*a2-3*a3|^2)/32 & r = (256*a0 -64*a3*a1 +16*a3|^2
*a2 -3*a3 |^4)/1024 & s1 = 2-root(p|^2-r) & 3_root_of_quartic(a0,a1,a2,a3) = 2
-root(-2*(p +s1))-a3/4)& ex p,r,s1 st p = (8*a2-3*a3|^2)/32 & r = (256*a0 -64*
a3*a1 +16*a3 |^2*a2 -3*a3|^4)/1024 & s1 = 2-root(p|^2-r) & 4_root_of_quartic(a0
    ,a1,a2,a3) = -2-root(-2*(p+s1))-a3/4 by Def7,Def8;
    ( ex p,r,s1 st p = (8*a2-3*a3|^2)/32 & r = (256*a0 -64*a3*a1 +16*a3|^2
*a2 -3*a3 |^4)/1024 & s1 = 2-root(p|^2-r) & 1_root_of_quartic(a0,a1,a2,a3) = 2
-root(-2*(p -s1))-a3/4)& ex p,r,s1 st p = (8*a2-3*a3|^2)/32 & r = (256*a0 -64*
a3*a1 +16*a3 |^2*a2 -3*a3|^4)/1024 & s1 = 2-root(p|^2-r) & 2_root_of_quartic(a0
    ,a1,a2,a3) = -2-root(-2*(p-s1))-a3/4 by A1,Def5,Def6;
    hence thesis by A2;
  end;
  suppose
A3: 8*a1 -4*a2*a3 + a3|^3 <> 0;
    then
A4: ( ex p,q,r,s1,s2,s3 st p = (8*a2-3*a3|^2)/32 & q = (8*a1 -4*a2*a3 + a3
    |^3)/64 & r = (256*a0 -64*a3*a1 +16*a3|^2*a2 -3*a3|^4)/1024 & s1 = 2-root(
1_root_of_cubic (-q|^2,p|^2-r,2*p)) & s2 = 2-root(2_root_of_cubic(-q|^2,p|^2-r,
2*p)) & s3 = -q/ (s1*s2) & 3_root_of_quartic(a0,a1,a2,a3) = -s1+s2-s3-a3/4)& ex
p,q,r,s1,s2,s3 st p = (8*a2-3*a3|^2)/32 & q = (8*a1 -4*a2*a3 + a3|^3)/64 & r =
(256*a0 -64*a3* a1 +16*a3|^2*a2 -3*a3|^4)/1024 & s1 = 2-root(1_root_of_cubic(-q
|^2,p|^2-r,2*p)) & s2 = 2-root(2_root_of_cubic(-q|^2,p|^2-r,2*p)) & s3 = -q/(s1
    *s2) & 4_root_of_quartic(a0,a1,a2,a3) = s1-s2-s3-a3/4 by Def7,Def8;
    ( ex p,q,r,s1,s2,s3 st p = (8*a2-3*a3|^2)/32 & q = (8*a1 -4*a2*a3 + a3
    |^3)/64 & r = (256*a0 -64*a3*a1 +16*a3|^2*a2 -3*a3|^4)/1024 & s1 = 2-root(
1_root_of_cubic (-q|^2,p|^2-r,2*p)) & s2 = 2-root(2_root_of_cubic(-q|^2,p|^2-r,
2*p)) & s3 = -q/ (s1*s2) & 1_root_of_quartic(a0,a1,a2,a3) = s1+s2+s3-a3/4)& ex
p,q,r,s1,s2,s3 st p = (8*a2-3*a3|^2)/32 & q = (8*a1 -4*a2*a3 + a3|^3)/64 & r =
(256*a0 -64*a3* a1 +16*a3|^2*a2 -3*a3|^4)/1024 & s1 = 2-root(1_root_of_cubic(-q
|^2,p|^2-r,2*p)) & s2 = 2-root(2_root_of_cubic(-q|^2,p|^2-r,2*p)) & s3 = -q/(s1
    *s2) & 2_root_of_quartic(a0,a1,a2,a3) = -s1-s2+s3-a3/4 by A3,Def5,Def6;
    hence thesis by A4;
  end;
end;
