 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 Th20:
  x is positive implies
    (L_||.x.|| \/ R_||.x.||)\{0_No} c= L_x \/R_x
proof
  assume
A1: x is positive;
  let a;
  assume
A2: a in (L_||.x.|| \/ R_||.x.||)\{0_No};
  then reconsider a as Surreal by SURREAL0:def 16;
A3:a in (L_||.x.|| \/ R_||.x.||) & a <> 0_No by A2,ZFMISC_1:56;
  a in L_||.x.|| or a in R_||.x.|| by A2,XBOOLE_0:def 3;
  then a in L_x or a in R_x by Def9,A1,A3;
  hence thesis by XBOOLE_0:def 3;
end;
