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=1 implies a*b+1/(a*b)>=17/4
proof
A1: sqrt(a*b)>0 by SQUARE_1:25;
  assume a+b=1;
  then 1^2>=(2*sqrt(a*b))^2 by A1,SIN_COS2:1,SQUARE_1:15;
  then 1>=2*2*(sqrt(a*b))^2;
  then 1>=4*(a*b) by SQUARE_1:def 2;
  then
A2: 1/4>=((a*b)*4)/4 by XREAL_1:72;
  then 1/4*(-1)<=a*b*(-1) by XREAL_1:65;
  then -(1/4)+1<=-(a*b)+1 by XREAL_1:7;
  then (3/4)^2<=(1-(a*b))^2 by SQUARE_1:15;
  then (9/4^2)/(1/4)<=((1-(a*b))^2)/(1/4) by XREAL_1:72;
  then
A3: 9/4+8/4<=((1-(a*b))^2)*4+2 by XREAL_1:7;
  (1-a*b)^2>=0 by XREAL_1:63;
  then (1-a*b)^2/(1/4)<=(1-a*b)^2/(a*b) by A2,XREAL_1:118;
  then
A4: (1-a*b)^2*4+2<=(1-a*b)^2/(a*b)+2 by XREAL_1:7;
  1/(a*b)+a*b =(1+(a*b)*(a*b))/(a*b) by XCMPLX_1:113
    .=((1^2-2*1*(a*b)+(a*b)^2)+2*(a*b))/(a*b)
    .=(1-a*b)^2/(a*b)+(2*(a*b))/(a*b) by XCMPLX_1:62
    .=(1-a*b)^2/(a*b)+2*((a*b)/(a*b)) by XCMPLX_1:74
    .=(1-a*b)^2/(a*b)+2*1 by XCMPLX_1:60
    .=(1-a*b)^2/(a*b)+2;
  hence thesis by A4,A3,XXREAL_0:2;
end;
