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
  sqrt(3*(a^2+b^2+c^2))<=(b*c)/a+(c*a)/b+(a*b)/c
proof
A1: ((b*c)/a)^2+((c*a)/b)^2>=2*c^2 by Lm13;
A2: ((c*a)/b)^2+((a*b)/c)^2>=2*a^2 by Lm13;
  ((b*c)/a)^2+((a*b)/c)^2>=2*b^2 by Lm13;
  then (((c*a)/b)^2+((a*b)/c)^2)+(((b*c)/a)^2+((a*b)/c)^2)>=2*a^2+2*b^2 by A2,
XREAL_1:7;
  then ((c*a)/b)^2+((a*b)/c)^2+((b*c)/a)^2+((a*b)/c)^2+(((b*c)/a)^2+((c*a)/b)
  ^2)>= 2*a^2+2*b^2+2*c^2 by A1,XREAL_1:7;
  then ((((c*a)/b)^2+((a*b)/c)^2+((b*c)/a)^2)*2)/2>=((a^2+b^2+c^2)*2)/2 by
XREAL_1:72;
  then
  ((c*a)/b)^2+((a*b)/c)^2+((b*c)/a)^2+2*(a^2+b^2+c^2)>=(a^2+b^2+c^2)+ 2*(a
  ^2+b^2+c^2) by XREAL_1:6;
  then ((b*c)/a+(c*a)/b+(a*b)/c)^2>=3*(a^2+b^2+c^2) by Lm16;
  then sqrt(((b*c)/a+(c*a)/b+(a*b)/c)^2)>=sqrt(3*(a^2+b^2+c^2)) by SQUARE_1:26;
  hence thesis by SQUARE_1:22;
end;
