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 (1+1/a)*(1+1/b)>=9
proof
  assume
A1: a+b=1;
  then 1/(a*b) >= 1/(1/4) by Th18,XREAL_1:85;
  then (1/(a*b))*(-2) <= 4*(-2) by XREAL_1:65;
  then -(1/(a*b))*2 <= -8;
  then (1/(a*b))*2 >= 8 by XREAL_1:24;
  then
A2: 1+2*(1/(a*b)) >= 1+8 by XREAL_1:7;
  (1+1/a)*(1+1/b)=1*1+1*(1/b)+(1/a)*(1/b)+(1/a)*1
    .=1+1/b+1/(a*b)+1/a by XCMPLX_1:102
    .=1+(1/b+1/a) +1/(a*b)
    .=1+(1*a+b*1)/(a*b)+1/(a*b) by XCMPLX_1:116
    .=1+1/(a*b)+1/(a*b) by A1
    .=1+2*(1/(a*b));
  hence thesis by A2;
end;
