 reserve A,B,O for Ordinal,
      n,m for Nat,
      a,b,o for object,
      x,y,z for Surreal,
      X,Y,Z for set,
      Inv,I1,I2 for Function;

theorem Th21:
  x is positive & y in (L_||.x.|| \/ R_||.x.||)\{0_No} implies y is positive
proof
  assume
A1: x is positive & y in (L_||.x.|| \/ R_||.x.||)\{0_No};
  then
A2:y in (L_||.x.|| \/ R_||.x.||) & y <> 0_No by ZFMISC_1:56;
  y in L_||.x.|| or y in R_||.x.|| by A1,XBOOLE_0:def 3;
  hence thesis by Def9,A1,A2;
end;
