reserve A,B for Ordinal,
        o for object,
        x,y,z for Surreal,
        n for Nat,
        r,r1,r2 for Real;

theorem
  x is positive &
       |. x - (No_omega^ y * uReal.r) .| infinitely< x
       implies r is positive
proof
  assume that
A1: x is positive & |. x - (No_omega^ y * uReal.r) .| infinitely< x and
A2: not r is positive;
A3:0_No <= x by A1;
A4:0_No <= No_omega^ y by SURREALI:def 8;
  uReal.r <= 0_No by A2,SURREALN:51,47;
  then No_omega^ y * uReal.r <= No_omega^ y * 0_No =0_No
  by A4,SURREALR:75;
  then -0_No <= - (No_omega^ y * uReal.r) by SURREALR:10;
  then
A5: x + 0_No <= x + - (No_omega^ y * uReal.r) &
  0_No + 0_No <= x + - (No_omega^ y * uReal.r)
  by A3,SURREALR:43;
  |. x +- (No_omega^ y * uReal.r) .| < x by A1,Th9;
  hence thesis by A5,Def6;
end;
