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

theorem Th4:
  0_No <= x implies 0_No <= NonNegativePart x
proof
  set NN=NonNegativePart x;
  assume
A1: 0_No <= x;
A2: L_0_No << {NN};
  {0_No} << R_NN
  proof
    let l,r be Surreal such that
A3: l in {0_No} & r in R_NN;
    consider R be Surreal such that
A4: R=r & R in R_x & 0_No <= R by A3,Def1;
    x in {x} << R_x by TARSKI:def 1,SURREALO:11;
    then 0_No < R by A4,A1,SURREALO:4;
    hence l < r by A3,A4,TARSKI:def 1;
  end;
  hence thesis by A2,SURREAL0:43;
end;
