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

theorem Th1:
  L_NonNegativePart o c= L_o & R_NonNegativePart o c= R_o
proof
  thus L_NonNegativePart o c= L_o
  proof
    let a be object;
    assume a in L_NonNegativePart o;
    then ex l be Surreal st a= l & l in L_o & 0_No <= l by Def1;
    hence thesis;
  end;
  let a be object;
  assume a in R_NonNegativePart o;
  then ex r be Surreal st a= r & r in R_o & 0_No <= r by Def1;
  hence thesis;
end;
