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+1/a)*(1+1/b)*(1+1/c)>=64
proof
A1: sqrt(b*c)>0 by SQUARE_1:25;
  (a+b)/c>=(2*sqrt(a*b))/c by SIN_COS2:1,XREAL_1:72;
  then 2+(a+b)/c>=2+(2*sqrt(a*b))/c by XREAL_1:6;
  then
A2: 2+(a+b)/c>=2+2*(sqrt(a*b)/c) by XCMPLX_1:74;
A3: sqrt(a*b)>0 by SQUARE_1:25;
  (a+c)/b>=(2*sqrt(a*c))/b by SIN_COS2:1,XREAL_1:72;
  then 2+(a+c)/b>=2+(2*sqrt(a*c))/b by XREAL_1:6;
  then
A4: 2+(a+c)/b>=2+2*(sqrt(a*c)/b) by XCMPLX_1:74;
A5: sqrt(a*c)>0 by SQUARE_1:25;
  (b+c)/a>=(2*sqrt(b*c))/a by SIN_COS2:1,XREAL_1:72;
  then 2+(b+c)/a>=2+(2*sqrt(b*c))/a by XREAL_1:6;
  then 2+(b+c)/a>=2+2*(sqrt(b*c)/a) by XCMPLX_1:74;
  then (2+(b+c)/a)*(2+(a+c)/b)>=(2*(1+sqrt(b*c)/a))*(2*(1+sqrt(a*c)/b)) by A4
,A1,A5,XREAL_1:66;
  then
A6: (2+(b+c)/a)*(2+(a+c)/b)*(2+(a+b)/c)>=(2*(1+sqrt(b*c)/a))*(2*(1+sqrt(a*
  c)/b)) *(2*(1+sqrt(a*b)/c)) by A2,A1,A5,A3,XREAL_1:66;
  sqrt(a*c)/b+b/sqrt(a*c)+sqrt(a*b)/c+c/sqrt(b*a)+a/sqrt(b*c)+sqrt(b*c)/a
  >=6 by Lm23;
  then 2+(sqrt(a*c)/b+b/sqrt(a*c)+sqrt(a*b)/c+c/sqrt(b*a)+a/sqrt(b*c)+sqrt(b*
  c)/a) >=6+2 by XREAL_1:6;
  then
A7: 8*(2+sqrt(a*c)/b+b/sqrt(a*c)+sqrt(a*b)/c+c/sqrt(b*a)+a/sqrt(b*c)+ sqrt(
  b*c)/a)>=8*8 by XREAL_1:64;
  assume
A8: a+b+c = 1;
  then (1+1/a)*(1+1/b)*(1+1/c) =(2+(b+c)/a)*(1+1/b)*(1+1/c) by Lm19
    .=(2+(b+c)/a)*(2+(a+c)/b)*(1+1/c) by A8,Lm19
    .=(2+(b+c)/a)*(2+(a+c)/b)*(1+((a+b)/c+c/c)) by A8,XCMPLX_1:62
    .=(2+(b+c)/a)*(2+(a+c)/b)*(1+(a+b)/c+1) by XCMPLX_1:60
    .=(2+(b+c)/a)*(2+(a+c)/b)*(2+(a+b)/c);
  then
  (1+1/a)*(1+1/b)*(1+1/c)>=8*((1+sqrt(b*c)/a)*(1+sqrt(a*c)/b) *(1+sqrt(a*
  b)/c)) by A6;
  then (1+1/a)*(1+1/b)*(1+1/c)>=8* (2+sqrt(a*c)/b+b/sqrt(a*c)+sqrt(a*b)/c+c/
  sqrt(b*a)+a/sqrt(b*c)+sqrt(b*c)/a) by Lm22;
  hence thesis by A7,XXREAL_0:2;
end;
