 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 Th24:
  x is positive implies
    (L_||.x.|| \/ R_||.x.||)\{0_No} c= Positives born x
proof
  assume A1:x is positive;
  then
A2:(L_||.x.|| \/ R_||.x.||)\{0_No} c= L_x \/ R_x by Th20;
  let a;
  assume
A3:a in (L_||.x.|| \/ R_||.x.||)\{0_No};
  then reconsider a as Surreal by SURREAL0:def 16;
A4:a is positive by A1,A3,Th21;
  born a in born x by A3,A2,SURREALO:1;
  then a in Day born a c= Day born x by SURREAL0:35,def 18,ORDINAL1:def 2;
  hence thesis by A4,Def10;
end;
