reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem
  a+b=1 implies (a+1/a)*(b+1/b)>=25/4
proof
  assume
A1: a+b=1;
  then
A2: a*b<=(1/2)^2 by Th4;
  then 1-(a*b)>=1-(1/4) by XREAL_1:10;
  then (1-(a*b))^2>=(3/4)^2 by SQUARE_1:15;
  then 1+(1-(a*b))^2>=1+(9/16) by XREAL_1:6;
  then
A3: 4*((1+((1-(a*b))^2)))>=4*(25/16) by XREAL_1:64;
  ((1-(a*b))^2)>=0 by XREAL_1:63;
  then
A4: (1+((1-(a*b))^2)) /(a*b)>=((1+((1-(a*b))^2)))/(1/4) by A2,XREAL_1:118;
  a^2+b^2=(a^2+2*a*b+b^2)-2*a*b;
  then
A5: a^2+b^2=1^2-2*a*b by A1,SQUARE_1:4;
  (a+(1/a))*(b+1/b)=((1+a^2)/a)*(b+1/b) by XCMPLX_1:113
    .=((1+a^2)/a)*((1+b^2)/b) by XCMPLX_1:113
    .=((1+a^2)*(1+b^2))/(a*b) by XCMPLX_1:76
    .=((1+(1^2-2*(a*b)+a^2*b^2)))/(a*b) by A5
    .=((1+((1-(a*b))^2)))/(a*b);
  hence thesis by A4,A3,XXREAL_0:2;
end;
