reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem
  x>0 & y>0 & z<0 & x+y+z=0 implies (x|^2+y|^2+z|^2)|^3>=6*(x|^3+y|^3+z |^3)^2
proof
  assume that
A1: x>0 & y>0 and
A2: z<0 and
A3: x+y+z=0;
  3 -Root(((x^2*y^2*z^2))/4)>0 by A1,A2,PREPOWER:def 2;
  then
A4: 3-root(((x^2*y^2*z^2))/4)>0 by A1,A2,POWER:def 1;
  x*(x+y)/2+y*(x+y)/2+x*y>=3*(3-root((x*(x+y)/2)*(y*(x+y)/2)*(x*y))) by A1,Th15
;
  then x*(x+y)/2+y*(x+y)/2+x*y>=3*(3-root(((x^2*y^2*((x+y)*(x+y))))/4));
  then
A5: x*(x+y)/2+y*(x+y)/2+x*y>=3*(3-root(((x^2*y^2*(-z)^2))/4)) by A3;
  (3*(3-root(((x^2*y^2*z^2))/4)))|^3 =3|^3*(3-root(((x^2*y^2*z^2))/4))|^3
  by NEWTON:7
    .=27*(3 -Root(((x^2*y^2*z^2))/4))|^3 by A1,A2,Lm3,POWER:def 1
    .=27*(((x^2*y^2*z^2))/4) by A1,A2,PREPOWER:19;
  then (x*(x+y)/2+y*(x+y)/2+x*y)|^3>=27*(((x^2*y^2*z^2))/4) by A5,A4,PREPOWER:9
;
  then
A6: 8*(x*(x+y)/2+y*(x+y)/2+x*y)|^3>=8*(27*(((x^2*y^2*z^2))/4)) by XREAL_1:64;
  (x^2+y^2)/2>=x*y by Th7;
  then (x|^2+y^2)/2>=x*y by Lm1;
  then (x|^2+y|^2)/2>=x*y by Lm1;
  then (x|^2+x*y)/2+(x*y+y|^2)/2+(x|^2+y|^2)/2>=(x|^2+x*y)/2+(x*y+y|^2)/2+x*y
  by XREAL_1:6;
  then (x|^2+x*y)/2+(x*y+y|^2)/2+(x|^2+y|^2)/2>=(x^2+x*y)/2+(x*y+y|^2)/2+x*y
  by Lm1;
  then
  (x|^2+x*y)/2+(x*y+y|^2)/2+(x|^2+y|^2)/2>=(x*x+x*y)/2+(x*y+y^2)/2+x*y by Lm1;
  then
  ((x|^2+x*y)/2+(x*y+y|^2)/2+(x|^2+y|^2)/2)|^3>=(x*(x+y)/2+y*(x+y)/2+x*y)
  |^3 by A1,PREPOWER:9;
  then
  8*((x|^2+x*y)/2+(x*y+y|^2)/2+(x|^2+y|^2)/2)|^3>=8*(x*(x+y)/2 +y*(x+y)/2
  +x*y)|^3 by XREAL_1:64;
  then 8*(x|^2+x*y+y|^2)|^3>=54*(x^2*y^2*z^2) by A6,XXREAL_0:2;
  then (2*(x|^2+x*y+y|^2))|^3>=54*(x^2*y^2*z^2) by Lm2,NEWTON:7;
  then ((x|^2+2*x*y+y|^2)+x|^2+y|^2)|^3>=54*(x^2*y^2*z^2);
  then ((x^2+2*x*y+y|^2)+x|^2+y|^2)|^3>=54*(x^2*y^2*z^2) by Lm1;
  then ((x^2+2*x*y+y|^2)+x^2+y|^2)|^3>=54*(x^2*y^2*z^2) by Lm1;
  then ((x^2+2*x*y+y^2)+x^2+y|^2)|^3>=54*(x^2*y^2*z^2) by Lm1;
  then
A7: ((x^2+2*x*y+y^2)+x^2+y^2)|^3>=54*(x^2*y^2*z^2) by Lm1;
A8: z=-(x+y) by A3;
  then z|^3 = -((x+y)|^3) by Lm4
    .=-(x|^3+3*x^2*y+3*x*y^2+y|^3) by Lm5;
  then (z|^2+x^2+y^2)|^3>=6*(x|^3+y|^3+z|^3)^2 by A8,A7,Lm1;
  then (z|^2+x|^2+y^2)|^3>=6*(x|^3+y|^3+z|^3)^2 by Lm1;
  then (z|^2+x|^2+y|^2)|^3>=6*(x|^3+y|^3+z|^3)^2 by Lm1;
  hence thesis;
end;
