 reserve n,m for Nat,
      o for object,
      p for pair object,
      x,y,z for Surreal;

theorem Th3:
  born NonNegativePart x c= born x
proof
  set NN = NonNegativePart x;
  for o be object st o in L_NN \/ R_NN
    ex O be Ordinal st O in born x & o in Day O
  proof
    let o be object such that
A1: o in L_NN \/ R_NN;
    reconsider o as Surreal by A1,SURREAL0:def 16;
    o in L_NN c= L_x or o in R_NN c= R_x by A1,Th1,XBOOLE_0:def 3;
    then o in L_x\/R_x by XBOOLE_0:def 3;
    then born o in born x & o in Day born o
    by SURREALO:1,SURREAL0:def 18;
    hence thesis;
  end;
  then [L_NN,R_NN] in Day born x by SURREAL0:45,SURREAL0:46;
  hence thesis by SURREAL0:def 18;
end;
