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 Th18:
  x+y=1 implies x*y<=1/4
proof
  (x-y)^2 >= 0 by XREAL_1:63;
  then
A1: x^2-2*x*y+y^2-(x^2+2*x*y+y^2)>=0-(x^2+2*x*y+y^2)by XREAL_1:9;
  assume x+y=1;
  then 1^2=x^2+2*x*y+y^2 by SQUARE_1:4;
  then -4*x*y>=-1 by A1;
  then 4*(x*y)<=1 by XREAL_1:24;
  then 4*(x*y)/4<=1/4 by XREAL_1:72;
  hence thesis;
end;
