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

theorem
  0_No <= x implies NonNegativePart x == x
proof
  set NN = NonNegativePart x;
  assume
A1: 0_No <= x;
  then
A2: 0_No <= NN by Th4;
A3: L_NN << {x}
  proof
    let l,r be Surreal such that
A4: l in L_NN & r in {x};
    L_NN c= L_x << {x} by Th1,SURREALO:11;
    hence thesis by A4;
  end;
A5: {NN} << R_x
  proof
    let l,r be Surreal such that
A6: l in {NN} & r in R_x;
    x in {x} << R_x by TARSKI:def 1,SURREALO:11;
    then 0_No <= r by A6,A1,SURREALO:4;
    then
A7: r in R_NN by A6,Def1;
    assume r <= l;
    then r <= NN by A6,TARSKI:def 1;
    then r in {r} << R_NN by TARSKI:def 1,SURREAL0:43;
    hence thesis by A7,SURREALO:3;
  end;
A8: L_x << {NN}
  proof
    let l,r be Surreal such that
A9: l in L_x & r in {NN};
    assume
A10:r <= l;
    then NN <= l by A9,TARSKI:def 1;
    then 0_No <= l by A2,SURREALO:4;
    then
A11:l in L_NN by A9,Def1;
    L_NN << {NN} by SURREALO:11;
    hence thesis by A10,A9,A11;
  end;
  {x} << R_NN
  proof
    let l,r be Surreal such that
A12: l in {x} & r in R_NN;
    R_NN c= R_x & {x} << R_x by Th1,SURREALO:11;
    hence thesis by A12;
  end;
  hence thesis by A5,A8, A3,SURREAL0:43;
end;
