reserve p,q,r,th,th1 for Real;
reserve n for Nat;

theorem Th1:
  p>=0 & r>=0 implies p+r>=2*sqrt(p*r)
proof
  assume that
A1: p>=0 and
A2: r>=0;
A3: (sqrt p - sqrt r)^2>=0 by XREAL_1:63;
  (sqrt p - sqrt r)^2 =(sqrt p)^2 - 2*sqrt p*sqrt r + (sqrt r)^2
    .=p-2*sqrt p*sqrt r + (sqrt r)^2 by A1,SQUARE_1:def 2
    .=p -2*sqrt p*sqrt r +r by A2,SQUARE_1:def 2
    .=p+r-2*sqrt p*sqrt r;
  then 0+2*sqrt p*sqrt r <= p+r by A3,XREAL_1:19;
  then 2*(sqrt p*sqrt r) <= p+r;
  hence thesis by A1,A2,SQUARE_1:29;
end;
