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 Th12:
  a|^3+b|^3+c|^3>=3*a*b*c
proof
A1: (a+c)|^3-(3*a^2*c+3*a*c^2)=a|^3+3*a^2*c+3*a*c^2+c|^3-(3*a^2*c+3*a*c^2)
  by Lm5;
  a*(b^2+c^2)>=a*(2*b*c) & b*(a^2+c^2)>=b*(2*a*c) by Th6,XREAL_1:64;
  then
A2: b*(a^2+c^2)+a*(b^2+c^2)>=2*a*b*c+2*a*b*c by XREAL_1:7;
  (a+c)^2*(a+c)>=4*a*c*(a+c) by Th9,XREAL_1:64;
  then (a+c)|^2*(a+c)>=4*a*c*(a+c) by Lm1;
  then
A3: (a+c)|^(2+1)>=4*a*c*(a+c) by NEWTON:6;
  (b+c)^2*(b+c)>=4*b*c*(b+c) by Th9,XREAL_1:64;
  then (b+c)|^2*(b+c)>=4*b*c*(b+c) by Lm1;
  then
A4: (b+c)|^(2+1)>=4*b*c*(b+c) by NEWTON:6;
  (a+b)^2*(a+b)>=4*a*b*(a+b) by Th9,XREAL_1:64;
  then (a+b)|^2*(a+b)>=4*a*b*(a+b) by Lm1;
  then (a+b)|^(2+1)>=4*a*b*(a+b) by NEWTON:6;
  then (a+b)|^3+(b+c)|^3>=(4*a^2*b+4*a*b^2)+(4*b^2*c+4*b*c^2) by A4,XREAL_1:7;
  then
  ((a+b)|^3+(b+c)|^3)+(a+c)|^3>=(4*a^2*b+4*a*b^2+4*b^2*c+4*b*c^2) +(4*a^2
  *c+4*a*c^2) by A3,XREAL_1:7;
  then
A5: (((a+b)|^3)+((b+c)|^3)+((a+c)|^3))+(-3*a^2*b-3*a*b^2-(3*b^2*c)-(3*b* c
^2 ) -(3*a^2*c)-(3*a*c^2))>=((4*a^2*b)+4*a*b^2+4*b^2*c+4*b*c^2+4*a^2*c+4*a*c^2)
  +(-3*a^2*b-3*a*b^2-(3*b^2*c)-(3*b*c^2)-(3*a^2*c)-(3*a*c^2)) by XREAL_1:6;
  c*(a^2+b^2)>=2*a*b*c by Th6,XREAL_1:64;
  then
A6: (b*(a^2+c^2)+a*(b^2+c^2))+(c*(a^2+b^2))>=4*a*b*c+2*a*b*c by A2,XREAL_1:7;
  (a+b)|^3-(3*a^2*b+3*a*b^2)=a|^3+3*a^2*b+3*a*b^2+b|^3-(3*a^2*b+3*a*b^2) &
(b+ c)|^3-(3*b^2*c+3*b*c^2)=b|^3+3*b^2*c+3*b*c^2+c|^3-(3*b^2*c+3*b*c^2) by Lm5;
  then 2*(a|^3+b|^3+(c|^3))>=6*a*b*c by A1,A5,A6,XXREAL_0:2;
  then (2*(a|^3+b|^3+(c|^3)))/2>=(6*a*b*c)/2 by XREAL_1:72;
  hence thesis;
end;
