reserve a,b,c,d for positive Real,
  m,u,w,x,y,z for Real,
  n,k for Nat,
  s,s1 for Real_Sequence;

theorem
  (9*a*b*c)/(a^2+b^2+c^2)<=a+b+c
proof
A1: b^2*(a+c)>=b^2*(2*sqrt(a*c)) by SIN_COS2:1,XREAL_1:64;
A2: c^2*(a+b)>=c^2*(2*sqrt(a*b)) by SIN_COS2:1,XREAL_1:64;
A3: sqrt(a*b)>0 by SQUARE_1:25;
A4: sqrt(a*c)>0 by SQUARE_1:25;
  sqrt(b*c)>0 by SQUARE_1:25;
  then
  2*a^2*sqrt(b*c)+2*b^2*sqrt(a*c)+2*c^2*sqrt(a*b)>= 3*(3-root((2*a^2*sqrt
  (b*c)) *(2*b^2*sqrt(a*c))*(2*c^2*sqrt(a*b)))) by A4,A3,SERIES_3:15;
  then
  2*a^2*sqrt(b*c)+2*b^2*sqrt(a*c)+2*c^2*sqrt(a*b)>= 3*(3-root(2*a^2*sqrt(
  b*c) *2*b^2*sqrt(a*c)*2*c^2*sqrt(a*b)));
  then
  2*a^2*sqrt(b*c)+2*b^2*sqrt(a*c)+2*c^2*sqrt(a*b)>= 3*(3-root((2*a*b*c)|^
  3)) by Lm9;
  then
A5: 2*a^2*sqrt(b*c)+2*b^2*sqrt(a*c)+2*c^2*sqrt(a*b)>= 3*(2*a*b*c) by POWER:4;
A6: a|^3+b|^3+c|^3>=3*a*b*c by SERIES_3:12;
  a^2*(b+c)>=a^2*(2*sqrt(b*c)) by SIN_COS2:1,XREAL_1:64;
  then a^2*(b+c)+b^2*(a+c)>=a^2*(2*sqrt(b*c))+b^2*(2*sqrt(a*c)) by A1,XREAL_1:7
;
  then a^2*(b+c)+b^2*(a+c)+c^2*(a+b)>=2*a^2*sqrt(b*c)+2*b^2*sqrt(a*c)+ 2*c^2*
  sqrt(a*b) by A2,XREAL_1:7;
  then a*b^2+a*c^2+b*a^2+b*c^2+(c*a^2)+(c*b^2)>=6*a*b*c by A5,XXREAL_0:2;
  then
A7: (a|^3+b|^3+c|^3)+(a*b^2+a*c^2+b*a^2+b*c^2+(c*a^2)+(c*b^2))>=3*a*b*c +6*
  a*b*c by A6,XREAL_1:7;
  (a^2+b^2+c^2)*(a+b+c) =a*a^2+a*b^2+a*c^2+b*a^2+b*b^2+b*c^2+(c*a^2)+(c*b
  ^2)+(c*c^2)
    .=a*a|^2+a*b^2+a*c^2+b*a^2+b*b^2+b*c^2+(c*a^2)+(c*b^2)+(c*c^2) by NEWTON:81
    .=a|^(2+1)+a*b^2+a*c^2+b*a^2+b*b^2+b*c^2+(c*a^2)+(c*b^2)+(c*c^2) by
NEWTON:6
    .=a|^3+a*b^2+a*c^2+b*a^2+b*b|^2+b*c^2+(c*a^2)+(c*b^2)+(c*c^2) by NEWTON:81
    .=a|^3+a*b^2+a*c^2+b*a^2+b|^(2+1)+b*c^2+(c*a^2)+(c*b^2)+(c*c^2) by NEWTON:6
    .=a|^3+a*b^2+a*c^2+b*a^2+b|^3+b*c^2+(c*a^2)+(c*b^2)+(c*c|^2) by NEWTON:81
    .=a|^3+a*b^2+a*c^2+b*a^2+b|^3+b*c^2+(c*a^2)+(c*b^2)+(c|^(2+1)) by NEWTON:6
    .=a|^3+b|^3+c|^3+a*b^2+a*c^2+b*a^2+b*c^2+(c*a^2)+(c*b^2);
  then ((a+b+c)*(a^2+b^2+c^2))/(a^2+b^2+c^2)>=(9*a*b*c)/(a^2+b^2+c^2) by A7,
XREAL_1:72;
  then (a+b+c)/((a^2+b^2+c^2)/(a^2+b^2+c^2))>=(9*a*b*c)/(a^2+b^2+c^2) by
XCMPLX_1:77;
  then (a+b+c)/1>=(9*a*b*c)/(a^2+b^2+c^2) by XCMPLX_1:60;
  hence thesis;
end;
