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
  (b*c)/a+(a*c)/b+(a*b)/c>=a+b+c
proof
A1: (a*c)/b+(a*b)/c = a*(c/b)+(a*b)/c by XCMPLX_1:74
    .=a*(c/b)+a*(b/c) by XCMPLX_1:74
    .=a*(c/b+b/c);
A2: (a*b)/c+(b*c)/a =b*(a/c)+(b*c)/a by XCMPLX_1:74
    .=b*(a/c)+b*(c/a) by XCMPLX_1:74
    .=b*(a/c+c/a);
A3: b*(a/c+c/a)>=b*2 by SERIES_3:3,XREAL_1:64;
A4: c*(b/a+a/b)>=c*2 by SERIES_3:3,XREAL_1:64;
  a*(c/b+b/c)>=a*2 by SERIES_3:3,XREAL_1:64;
  then a*(c/b+b/c)+b*(a/c+c/a)>=a*2+b*2 by A3,XREAL_1:7;
  then
A5: a*(c/b+b/c)+b*(a/c+c/a)+c*(b/a+a/b)>=a*2+b*2+c*2 by A4,XREAL_1:7;
  (b*c)/a+(a*c)/b = c*(b/a)+(a*c)/b by XCMPLX_1:74
    .=c*(b/a)+c*(a/b) by XCMPLX_1:74
    .=c*(b/a+a/b);
  then (2*((b*c)/a+(a*c)/b+(a*b)/c))/2>=(2*(a+b+c))/2 by A1,A2,A5,XREAL_1:72;
  hence thesis;
end;
