reserve a, a9, a1, a2, a3, b, b9, c, c9, d, d9, h, p, q, x, x1, x2, x3, u, v,
  y, z for Real;

theorem Th19:
  Polynom(1,0,p,q,y) = 0 implies for u,v st y = u+v & 3*v*u+p = 0
holds y = 3-root(-q/2+sqrt(q^2/4+(p/3) |^ 3)) + 3-root(-q/2-sqrt(q^2/4+(p/3) |^
  3)) or y = 3-root(-q/2+sqrt(q^2/4+(p/3) |^ 3)) + 3-root(-q/2+sqrt(q^2/4+(p/3)
|^ 3)) or y = 3-root(-q/2-sqrt(q^2/4+(p/3) |^ 3)) + 3-root(-q/2-sqrt(q^2/4+(p/3
  ) |^ 3))
proof
  set e1 = 1;
  assume
A1: Polynom(1,0,p,q,y) = 0;
  set e3 = (-p/3) |^ 3;
  set e2 = q;
  let u,v;
  assume that
A2: y = u+v and
A3: 3*v*u+p = 0;
  set z2 = v |^ 3;
  set z1 = u |^ 3;
A4: now
    let z;
    thus (z-z1)*(z-z2)= z^2-(z1+z2)*z+z1*z2
      .= z^2-(-q)*z+z1*z2 by A1,A2,A3,Th18
      .= z^2+q*z+(-p/3) |^ 3 by A1,A2,A3,Th18;
  end;
A5: z1+ z2 = -q by A1,A2,A3,Th18;
  then e2^2 = (z1+z2)^2 by SQUARE_1:3;
  then
A6: e2^2-(4*e1*e3) = -(-(z1^2+2*z1*z2+z2^2))-4*(z1*z2) by A1,A2,A3,Th18
    .= (z1-z2)^2;
  then
A7: (e2^2-4*e1*e3)>= 0 by XREAL_1:63;
  then
A8: delta(e1,e2,e3) >= 0 by QUIN_1:def 1;
  (z1-z1)*(z1-z2)= 0*(z1-z2);
  then
A9: Polynom(e1,e2,e3,z1) = 0 by A4;
  (z2-z1)*(z2-z2) = (z2-z1)*0;
  then
A10: Polynom(e1,e2,e3,z2) = 0 by A4;
A11: z2*z1 = (-p/3) |^ 3 by A1,A2,A3,Th18;
  now
    per cases by A9,A8,Th5;
    case
A12:  z1 = (-e2+sqrt delta(e1,e2,e3))/(2*e1);
A13:  (p/3) |^ 3 = (p/3) |^ (2+1) .= ((p/3) |^ (1+1))*(p/3) by NEWTON:6
        .= ((((p/3) |^ 1)*(p/3)))*(p/3) by NEWTON:6
        .= ((p/3) |^ 1)*(p/3)^2
        .= ((p/3) to_power 1)*(p/3)^2by POWER:41
        .= (p/3)*(p/3)^2 by POWER:25;
A14:  (q^2-4*(-p/3) |^ 3)>= 0 by A6,XREAL_1:63;
A15:  (-p/3) |^ 3 = (-p/3) |^ (2+1) .= ((-p/3) |^ (1+1))*(-p/3) by NEWTON:6
        .= ((((-p/3) |^ 1)*(-p/3)))*(-p/3) by NEWTON:6
        .= ((-p/3) |^ 1)*(-p/3)^2;
      then
A16:  (-p/3) |^ 3 = ((-p/3) to_power 1)*(-p/3)^2 by POWER:41
        .= (-p/3)*(p/3)^2 by POWER:25
        .= -((p/3)*(p/3)^2);
A17:  z1 = (-e2+sqrt(e2^2-4*e1*e3))/(2*e1) by A12,QUIN_1:def 1
        .= (-q)/2 +sqrt(q^2-4*(-p/3) |^ 3)/sqrt 4 by SQUARE_1:20,XCMPLX_1:62
        .= (-q)/2 +sqrt((q^2-4*(-p/3) |^ 3)/4) by A14,SQUARE_1:30
        .= (-q)/2 +sqrt(q^2/4-1*(-p/3) |^ 3);
A18:  now
        per cases by XXREAL_0:1;
        case
A19:      u >0;
          then
A20:      (-q/2 +sqrt(q^2/4+(p/3) |^ 3))> 0 by A17,A16,A13,PREPOWER:6;
          then u = 3 -Root (-q/2 +sqrt(q^2/4+(p/3) |^ 3)) by A17,A16,A13,A19,
PREPOWER:def 2;
          hence u = 3-root(-q/2 +sqrt(q^2 /4+(p/3) |^ 3)) by A20,POWER:def 1;
        end;
        case
A21:      u =0;
          then
A22:      -q/2 +sqrt(q^2/4+(p/3) |^ 3) = 0 by A17,A16,A13,NEWTON:11;
          then 3 -Root (-q/2 +sqrt(q^2/4+(p/3) |^ 3)) = 0 by PREPOWER:def 2;
          hence u = 3-root(-q/2 +sqrt(q^2/4+(p/3) |^ 3)) by A21,A22,POWER:def 1
;
        end;
        case
          u <0;
          then
A23:      -u > 0 by XREAL_1:58;
          set r = (-q/2 +sqrt(q^2/4+(p/3) |^ 3));
A24:      (3-root (-1)) = -1 by Lm1,POWER:8;
          (-u) |^ 3 = (-u) |^ (2+1) .= ((-u) |^ (1+1))*(-u) by NEWTON:6
            .= ((((-u) |^ 1)*(-u)))*(-u) by NEWTON:6
            .= ((-u) |^ 1)*(-u)^2;
          then (-u) |^ 3 = ((-u) to_power 1)*(-u)^2by POWER:41;
          then
A25:      (-u) |^ 3 = (-u)*u^2 by POWER:25
            .= -((u)*u^2);
          u |^ 3 = u |^ (2+1) .= (u |^ (1+1))*u by NEWTON:6
            .= ((u |^ 1)*u)*u by NEWTON:6
            .= (u |^ 1)*(u*u);
          then
A26:      u |^ 3 = (u to_power 1)*u^2 by POWER:41;
          then
A27:      (-u) |^ 3 = -(-q/2 +sqrt(q^2/4+(p/3) |^ 3)) by A17,A16,A13,A25,
POWER:25;
A28:      (-u) |^ 3 = -(u |^ 3) by A25,A26,POWER:25;
          then
A29:      -(-q/2 +sqrt(q^2/4+(p/3) |^ 3))> 0 by A17,A16,A13,A23,PREPOWER:6;
          -(u |^ 3)> 0 by A23,A28,PREPOWER:6;
          then
          (-u) = 3 -Root(-(-q/2 +sqrt(q^2/4+(p/3) |^ 3))) by A17,A16,A13,A23
,A27,PREPOWER:def 2;
          then (-u) = 3-root((-1)*(r)) by A29,POWER:def 1;
          then (-u) = (3-root (-1))*(3-root r) by Lm1,POWER:11;
          hence u = 3-root(-q/2 +sqrt ( q^2/4+(p/3) |^ 3)) by A24;
        end;
      end;
      now
        per cases by A8,A10,Th5;
        case
          z2 = (-e2+sqrt delta(e1,e2,e3))/(2*e1);
          then z2 = (-e2+sqrt(e2^2-4*e1*e3))/(2*e1) by QUIN_1:def 1;
          then z2 = (-q)/2 +sqrt(q^2-4*(-p/3) |^ 3)/sqrt 4 by SQUARE_1:20
,XCMPLX_1:62;
          then
A30:      z2 = (-q)/2 +sqrt((q^2-4*(-p/3) |^ 3)/4) by A7,SQUARE_1:30
            .= (-q)/2 +sqrt(q^2/4-1*(-p/3) |^ 3);
A31:      (-p/3) |^ 3 = ((-p/3) to_power 1)*(-p/3)^2by A15,POWER:41
            .= (-p/3)*(p/3)^2 by POWER:25;
          now
            per cases by XXREAL_0:1;
            case
A32:          v >0;
              then
A33:          (-q/2 +sqrt(q^2/4+(p/3) |^ 3))> 0 by A13,A30,A31,PREPOWER:6;
              then v = 3 -Root (-q/2 +sqrt(q^2/4+(p/3) |^ 3)) by A13,A30,A31
,A32,PREPOWER:def 2;
              hence v = 3-root(-q/2 +sqrt(q^2/4+(p/3) |^ 3)) by A33,POWER:def 1
;
            end;
            case
A34:          v=0;
              then (-q/2 +sqrt(q^2/4+(p/3) |^ 3)) = 0 by A13,A30,A31,NEWTON:11;
              hence v = 3-root(-q/2 +sqrt( q^2/4+(p/3) |^ 3)) by A34,POWER:5;
            end;
            case
              v<0;
              then
A35:          -v > 0 by XREAL_1:58;
              then
A36:          (-v) |^ 3 > 0 by PREPOWER:6;
              (-v) |^ 3 = (-v) |^ (2+1);
              then (-v) |^ 3 = ((-v) |^ (1+1))*(-v) by NEWTON:6;
              then (-v) |^ 3 = ((((-v) |^ 1)*(-v)))*(-v) by NEWTON:6;
              then (-v) |^ 3 = ((-v) |^ 1)*((-v)*(-v));
              then (-v) |^ 3 = ((-v) to_power 1)*(-v)^2by POWER:41;
              then (-v) |^ 3 = (-v)*(-v)^2 by POWER:25;
              then
A37:          (-v) |^ 3 = -(v*v^2);
              v |^ 3 = v |^ (2+1);
              then v |^ 3 = (v |^ (1+1))*v by NEWTON:6;
              then v |^ 3 = ((v |^ 1)*v)*v by NEWTON:6;
              then v |^ 3 = (v |^ 1)*(v*v);
              then
A38:          v |^ 3 = (v to_power 1)*v^2 by POWER:41;
              then
              (-v) |^ 3 = -(-q/2 +sqrt(q^2/4+(p/3) |^ 3)) by A13,A30,A31,A37,
POWER:25;
              then
A39:          (-v) = 3 -Root(-(-q/2 +sqrt(q^2/4+(p/3) |^ 3)) ) by A35,A36,
PREPOWER:def 2;
              set r = (-q/2 +sqrt(q^2/4+(p/3) |^ 3));
A40:          (3-root (-1)) = -1 by Lm1,POWER:8;
              -(-q/2 +sqrt(q^2/4+(p/3) |^ 3))> 0 by A13,A30,A31,A36,A37,A38,
POWER:25;
              then (-v) = 3-root((-1)*(r)) by A39,POWER:def 1;
              then (-v) = (3-root(-1))*(3-root r) by Lm1,POWER:11;
              hence v = 3-root(-q/2 +sqrt(q^2/4+(p/3) |^ 3)) by A40;
            end;
          end;
          hence thesis by A2,A18;
        end;
        case
          z2 = (-e2-sqrt delta(e1,e2,e3))/(2*e1);
          then z2 = (-e2-sqrt(e2^2-4*e1*e3))/(2*e1) by QUIN_1:def 1;
          then z2 = (-q)/2 -(sqrt(q^2-4*(-p/3) |^ 3))/2;
          then
A41:      z2 = -q/2 -sqrt((q^2-4*(-p/3) |^ 3)/4) by A7,SQUARE_1:20,30
            .= -q/2 -sqrt(q^2/4-1*(-p/3) |^ 3);
          now
            per cases by XXREAL_0:1;
            case
A42:          v>0;
              then (-q/2 -sqrt(q^2/4+(p/3) |^ 3))> 0 by A16,A13,A41,PREPOWER:6;
              then
A43:          v = 3 -Root (-q/2 -sqrt(q^2/4+(p/3) |^ 3)) by A16,A13,A41,A42,
PREPOWER:def 2;
              (-q/2 -sqrt(q^2/4+(p/3) |^ 3))> 0 by A16,A13,A41,A42,PREPOWER:6;
              hence v = 3-root (-q/2 -sqrt(q^2/4+(p/3) |^ 3)) by A43,
POWER:def 1;
            end;
            case
A44:          v=0;
              then
A45:          (-q/2 -sqrt(q^2/4+(p/3) |^ 3)) = 0 by A16,A13,A41,NEWTON:11;
              hence v = 3 -Root (-q/2 -sqrt (q^2/4+(p/3) |^ 3)) by A44,
PREPOWER:def 2;
              hence v = 3-root (-q/2 -sqrt(q ^2 /4+(p/3) |^ 3))by A45,
POWER:def 1;
            end;
            case
              v<0;
              then
A46:          -v > 0 by XREAL_1:58;
              set r = (-q/2 -sqrt(q^2/4+(p/3) |^ 3));
A47:          (3-root (-1)) = -1 by Lm1,POWER:8;
              v |^ 3 = v |^ (2+1);
              then v |^ 3 = (v |^ (1+1))*v by NEWTON:6;
              then
A48:          v |^ 3 = ((v |^ 1)*v)*v by NEWTON:6;
              (-v) |^ 3 = (-v) |^ (2+1);
              then (-v) |^ 3 = ((-v) |^ (1+1))*(-v) by NEWTON:6;
              then (-v) |^ 3 = ((((-v) |^ 1)*(-v)))*(-v) by NEWTON:6;
              then (-v) |^ 3 = ((-v) |^ 1)*((-v)*(-v));
              then
A49:          (-v) |^ 3 = ((-v) to_power 1)*(-v)^2 by POWER:41
                .= (-v)*v^2 by POWER:25
                .= -(v*v^2);
              (-v) |^ 3 = -(v |^ 3) by A49,A48;
              then
A50:          -(-q/2 -sqrt(q^2/4+(p/3) |^ 3))> 0 by A16,A13,A41,A46,PREPOWER:6;
              (-v) |^ 3 = -(-q/2 -sqrt(q^2/4+(p/3) |^ 3)) by A16,A13,A41,A49
,A48;
              then (-v) = 3 -Root(-(-q/2 -sqrt(q^2/4+(p/3) |^ 3)) ) by A46,A50,
PREPOWER:def 2;
              then (-v) = 3-root((-1)*(r)) by A50,POWER:def 1;
              then (-v) = (3-root (-1))*(3-root r) by Lm1,POWER:11;
              hence v = 3-root(-q/2 -sqrt(q^2/4+(p/3) |^ 3)) by A47;
            end;
          end;
          hence thesis by A2,A18;
        end;
      end;
      hence thesis;
    end;
    case
A51:  z1 = (-e2-sqrt delta(e1,e2,e3))/(2*e1);
      (-p/3) |^ 3 = (-p/3) |^ (2+1);
      then (-p/3) |^ 3 = ((-p/3) |^ (1+1))*(-p/3) by NEWTON:6;
      then (-p/3) |^ 3 = ((((-p/3) |^ 1)*(-p/3)))*(-p/3) by NEWTON:6;
      then (-p/3) |^ 3 = ((-p/3) |^ 1)*((-p/3)*(-p/3));
      then (-p/3) |^ 3 = ((-p/3) to_power 1)*(-p/3)^2 by POWER:41;
      then (-p/3) |^ 3 = (-p/3)*(-p/3)^2 by POWER:25;
      then
A52:  (-p/3) |^ 3 = -((p/3)*(p/3)^2);
      z1 = (-e2-sqrt(e2^2-4*e1*e3))/(2*e1) by A51,QUIN_1:def 1;
      then
A53:  z1= ((-q)*2") -(sqrt(q^2-4*(-p/3) |^ 3))/2
        .= -q/2 -sqrt((q^2-4*(-p/3) |^ 3)/4) by A7,SQUARE_1:20,30
        .= -q/2 -sqrt(q^2/4-1*(-p/3) |^ 3);
      hence z1 = -q/2 -sqrt(q^2/4-(-p/3) |^ 3);
      (p/3) |^ 3 = (p/3) |^ (2+1);
      then (p/3) |^ 3 = ((p/3) |^ (1+1))*(p/3) by NEWTON:6;
      then (p/3) |^ 3 = ((((p/3) |^ 1)*(p/3)))*(p/3) by NEWTON:6;
      then (p/3) |^ 3 = ((p/3) |^ 1)*((p/3)*(p/3));
      then (p/3) |^ 3 = ((p/3) to_power 1)*(p/3)^2by POWER:41;
      then
A54:  (-p/3) |^ 3 = -((p/3) |^ 3) by A52,POWER:25;
A55:  now
        per cases by XXREAL_0:1;
        case
A56:      u >0;
          then (-q/2 -sqrt(q^2/4+(p/3) |^ 3))> 0 by A53,A54,PREPOWER:6;
          then
A57:      u = 3 -Root (-q/2 -sqrt(q^2/4+(p/3) |^ 3)) by A53,A54,A56,
PREPOWER:def 2;
          (-q/2 -sqrt(q^2 /4+(p/3) |^ 3))> 0 by A53,A54,A56,PREPOWER:6;
          hence u = 3-root (-q/2 - sqrt(q^2/4+(p/3) |^ 3)) by A57,POWER:def 1;
        end;
        case
A58:      u =0;
          then (-q/2 -sqrt(q^2/4+(p/3) |^ 3)) = 0 by A53,A54,NEWTON:11;
          hence u = 3-root(-q/2 -sqrt(q^2/4+(p/3) |^ 3)) by A58,POWER:5;
        end;
        case
          u <0;
          then
A59:      -u > 0 by XREAL_1:58;
          then
A60:      (-u) |^ 3 > 0 by PREPOWER:6;
          (-u) |^ 3 = (-u) |^ (2+1);
          then (-u) |^ 3 = ((-u) |^ (1+1))*(-u) by NEWTON:6;
          then (-u) |^ 3 = ((((-u) |^ 1)*(-u)))*(-u) by NEWTON:6;
          then (-u) |^ 3 = ((-u) |^ 1)*((-u)*(-u));
          then (-u) |^ 3 = ((-u) to_power 1)*(-u)^2 by POWER:41;
          then (-u) |^ 3 = (-u)*(-u)^2 by POWER:25;
          then
A61:      (-u) |^ 3 = -(u*u^2);
          u |^ 3 = u |^ (2+1);
          then u |^ 3 = (u |^ (1+1))*u by NEWTON:6;
          then u |^ 3 = ((u |^ 1)*u)*u by NEWTON:6;
          then u |^ 3 = (u |^ 1)*(u*u);
          then
A62:      u |^ 3 = (u to_power 1)*u^2 by POWER:41;
          then -(-q/2 -sqrt(q^2/4+(p/3) |^ 3))> 0 by A53,A54,A60,A61,POWER:25;
          then
A63:      3 -Root(-(-q/2 -sqrt(q^2/4+(p/3) |^ 3))) = 3-root(-(-q/2 -sqrt
          (q^2/4+(p/3) |^ 3))) by POWER:def 1;
          set r = (-q/2 -sqrt(q^2/4+(p/3) |^ 3));
          (-u) |^ 3 = -(-q/2 -sqrt(q^2/4+(p/3) |^ 3)) by A53,A54,A61,A62,
POWER:25;
          then (-u) = 3-root((-1)*(r)) by A59,A60,A63,PREPOWER:def 2;
          then
A64:      (-u) = (3-root (-1))*(3-root r) by Lm1,POWER:11;
          (3-root (-1)) = -1 by Lm1,POWER:8;
          hence u = 3-root(-q / 2 -sqrt(q^2/4+(p/3) |^ 3)) by A64;
        end;
      end;
      now
        per cases by A8,A10,Th5;
        case
A65:      z2 = (-e2+sqrt delta(e1,e2,e3))/(2*e1);
          (-p/3) |^ 3 = (-p/3) |^ (2+1);
          then (-p/3) |^ 3 = ((-p/3) |^ (1+1))*(-p/3) by NEWTON:6;
          then (-p/3) |^ 3 = ((((-p/3) |^ 1)*(-p/3)))*(-p/3) by NEWTON:6;
          then (-p/3) |^ 3 = ((-p/3) |^ 1)*((-p/3)*(-p/3));
          then (-p/3) |^ 3 = ((-p/3) to_power 1)*(-p/3)^2by POWER:41;
          then (-p/3) |^ 3 = (-p/3)*(-p/3)^2 by POWER:25;
          then
A66:      (-p/3) |^ 3 = -((p/3)*(p/3)^2);
          (p/3) |^ 3 = (p/3) |^ (2+1);
          then (p/3) |^ 3 = ((p/3) |^ (1+1))*(p/3) by NEWTON:6;
          then (p/3) |^ 3 = ((((p/3) |^ 1)*(p/3)))*(p/3) by NEWTON:6;
          then (p/3) |^ 3 = ((p/3) |^ 1)*((p/3)*(p/3));
          then (p/3) |^ 3 = ((p/3) to_power 1)*(p/3)^2by POWER:41;
          then
A67:      (-p/3) |^ 3 = -((p/3) |^ 3) by A66,POWER:25;
          z2 = (-e2+sqrt(e2^2-4*e1*e3))/(2*e1) by A65,QUIN_1:def 1;
          then z2 = (-q)/2 +sqrt(q^2-4*(-p/3) |^ 3)/sqrt 4 by SQUARE_1:20
,XCMPLX_1:62;
          then
A68:      z2 = (-q)/2 +sqrt((q^2-4*(-p/3) |^ 3)/4) by A7,SQUARE_1:30
            .= (-q)/2 +sqrt(q^2/4-1*(-p/3) |^ 3);
          now
            per cases by XXREAL_0:1;
            case
A69:          v >0;
              then
A70:          (-q/2 +sqrt(q^2/4+(p/3) |^ 3))> 0 by A68,A67,PREPOWER:6;
              then v = 3 -Root (-q/2 +sqrt(q^2/4+(p/3) |^ 3)) by A68,A67,A69,
PREPOWER:def 2;
              hence v = 3-root(-q/2 + sqrt(q^2/4+(p/3) |^ 3)) by A70,
POWER:def 1;
            end;
            case
A71:          v=0;
              then (-q/2 +sqrt(q^2/4+(p/3) |^ 3)) = 0 by A68,A67,NEWTON:11;
              hence v = 3-root(-q/2 +sqrt(q^2/ 4+(p/3) |^ 3)) by A71,POWER:5;
            end;
            case
              v<0;
              then
A72:          -v > 0 by XREAL_1:58;
              then
A73:          (-v) |^ 3 > 0 by PREPOWER:6;
              (-v) |^ 3 = (-v) |^ (2+1);
              then (-v) |^ 3 = ((-v) |^ (1+1))*(-v) by NEWTON:6;
              then (-v) |^ 3 = ((((-v) |^ 1)*(-v)))*(-v) by NEWTON:6;
              then (-v) |^ 3 = ((-v) |^ 1)*((-v)*(-v));
              then (-v) |^ 3 = ((-v) to_power 1)*(-v)^2by POWER:41;
              then (-v) |^ 3 = (-v)*(-v)^2 by POWER:25;
              then
A74:          (-v) |^ 3 = -(v*v^2);
              v |^ 3 = v |^ (2+1);
              then v |^ 3 = (v |^ (1+1))*v by NEWTON:6;
              then v |^ 3 = ((v |^ 1)*v)*v by NEWTON:6;
              then v |^ 3 = (v |^ 1)*(v*v);
              then
A75:          v |^ 3 = (v to_power 1)*v^2 by POWER:41;
              then
A76:          -(-q/2 +sqrt(q^2/4+(p/3) |^ 3))> 0 by A68,A67,A73,A74,POWER:25;
              set r = (-q/2 +sqrt(q^2/4+(p/3) |^ 3));
A77:          (3-root (-1)) = -1 by Lm1,POWER:8;
              v |^ 3 = v*v^2 by A75,POWER:25;
              then (-v) = 3 -Root(-(-q/2 +sqrt(q^2/4+(p/3) |^ 3))) by A68,A67
,A72,A73,A74,PREPOWER:def 2;
              then (-v) = 3-root((-1)*(r)) by A76,POWER:def 1;
              then (-v) = (3-root(-1))*(3-root r) by Lm1,POWER:11;
              hence v = 3-root(-q/2 +sqrt(q^2/4+(p/3) |^ 3)) by A77;
            end;
          end;
          hence thesis by A2,A55;
        end;
        case
A78:      z2 = (-e2-sqrt delta(e1,e2,e3))/(2*e1);
          e2^2 = (z1+z2)^2 by A5,SQUARE_1:3;
          then
A79:      e2^2-(4*e1*e3) = (z1-z2)^2 by A11;
          (-p/3) |^ 3 = (-p/3) |^ (2+1)
            .= ((-p/3) |^ (1+1))*(-p/3) by NEWTON:6
            .= ((((-p/3) |^ 1)*(-p/3)))*(-p/3) by NEWTON:6
            .= ((-p/3) |^ 1)*((-p/3)*(-p/3));
          then (-p/3) |^ 3 = ((-p/3) to_power 1)*(-p/3)^2by POWER:41
            .= (-p/3)*(p/3)^2 by POWER:25;
          then
A80:      (-p/3) |^ 3 = -((p/3)*(p/3)^2);
          (p/3) |^ 3 = (p/3) |^ (2+1) .= ((p/3) |^ (1+1))*(p/3) by NEWTON:6
            .= ((((p/3) |^ 1)*(p/3)))*(p/3) by NEWTON:6
            .= ((p/3) |^ 1)*(p/3)^2;
          then (p/3) |^ 3 = ((p/3) to_power 1)*(p/3)^2by POWER:41;
          then
A81:      (-p/3) |^ 3 = -((p/3) |^ 3) by A80,POWER:25;
          z2 = (-e2-sqrt(e2^2-4*e1*e3))/(2*e1) by A78,QUIN_1:def 1;
          then
A82:      z2 = (-q)/2 +sqrt((q^2-4*(-p/3) |^ 3)/4) by A51,A78,A79,SQUARE_1:17
            .= (-q)/2 +sqrt(q^2/4-1*(-p/3) |^ 3);
          now
            per cases by XXREAL_0:1;
            case
A83:          v >0;
              then
A84:          (-q/2 +sqrt(q^2/4+(p/3) |^ 3))> 0 by A82,A81,PREPOWER:6;
              then v = 3 -Root (-q/2 +sqrt(q^2/4+(p/3) |^ 3)) by A82,A81,A83,
PREPOWER:def 2;
              hence v = 3-root(-q/2 +sqrt (q^2/4+(p/3) |^ 3)) by A84,
POWER:def 1;
            end;
            case
A85:          v=0;
              then v |^ 3 = 0 by NEWTON:11;
              hence
              v = 3-root(-q/2 +sqrt(q ^2/4+(p/3) |^ 3)) by A82,A81,A85,POWER:5;
            end;
            case
              v<0;
              then
A86:          -v > 0 by XREAL_1:58;
              then
A87:          (-v) |^ 3 > 0 by PREPOWER:6;
              set r = (-q/2 +sqrt(q^2/4+(p/3) |^ 3));
              v |^ 3 = v |^ (2+1) .= (v |^ (1+1))*v by NEWTON:6
                .= ((v |^ 1)*v)*v by NEWTON:6
                .= (v |^ 1)*v^2;
              then
A88:          v |^ 3 = (v to_power 1)*v^2 by POWER:41;
              (-v) |^ 3 = (-v) |^ (2+1) .= ((-v) |^ (1+1))*(-v) by NEWTON:6
                .= ((((-v) |^ 1)*(-v)))*(-v) by NEWTON:6
                .= ((-v) |^ 1)*(-v)^2
                .= ((-v) to_power 1)*(-v)^2by POWER:41
                .= (-v)*v^2 by POWER:25;
              then
A89:          (-v) |^ 3 = -(v*v^2);
              then
A90:          -(v |^ 3)> 0 by A87,A88,POWER:25;
A91:          (3-root (-1)) = -1 by Lm1,POWER:8;
              (-v) |^ 3 = -(v |^ 3) by A89,A88,POWER:25;
              then (-v) = 3 -Root(-(-q/2 +sqrt(q^2/4+(p/3) |^ 3))) by A82,A81
,A86,A87,PREPOWER:def 2;
              then (-v) = 3-root((-1)*(r)) by A82,A81,A90,POWER:def 1;
              then (-v) = (3-root(-1))*(3-root r) by Lm1,POWER:11;
              hence v = 3-root(-q/2 +sqrt(q^2/4+(p/3) |^ 3)) by A91;
            end;
          end;
          hence
          y = 3-root(-q/2 +sqrt(q^2/4+(p/3) |^ 3)) +3-root(-q/2 -sqrt(q^2
          /4+(p/3) |^ 3)) by A2,A55;
        end;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
