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
  a+b+c = 1 implies (1/a-1)*(1/b-1)*(1/c-1)>=8
proof
A1: (a+c)/b>=(2*sqrt(a*c))/b by SIN_COS2:1,XREAL_1:72;
A2: (a+b)/c>=(2*sqrt(a*b))/c by SIN_COS2:1,XREAL_1:72;
A3: sqrt(a*b)>0 by SQUARE_1:25;
A4: sqrt(a*c)>0 by SQUARE_1:25;
  assume
A5: a+b+c =1;
  then
A6: (1/a-1)*(1/b-1)*(1/c-1) =((b+c)/a)*(1/b-1)*(1/c-1) by Lm17
    .=((b+c)/a)*((a+c)/b)*(1/c-1) by A5,Lm17
    .=((b+c)/a)*((a+c)/b)*((a+b)/c+c/c-1) by A5,XCMPLX_1:62
    .=((b+c)/a)*((a+c)/b)*((a+b)/c+1-1) by XCMPLX_1:60
    .=((b+c)/a)*((a+c)/b)*((a+b)/c);
A7: sqrt(b*c)>0 by SQUARE_1:25;
  (b+c)/a>=(2*sqrt(b*c))/a by SIN_COS2:1,XREAL_1:72;
  then ((b+c)/a)*((a+c)/b)>=((2*sqrt(b*c))/a)*((2*sqrt(a*c))/b) by A1,A7,A4,
XREAL_1:66;
  then
  (1/a-1)*(1/b-1)*(1/c-1)>=((2*sqrt(b*c))/a)*((2*sqrt(a*c))/b)* ((2*sqrt(
  a*b))/c) by A6,A2,A7,A4,A3,XREAL_1:66;
  hence thesis by Lm18;
end;
