theorem Th2:
  (x in L_NonNegativePart o iff x in L_o & 0_No <= x) &
  (x in R_NonNegativePart o iff x in R_o & 0_No <= x)
proof
A1: x in L_NonNegativePart o implies x in L_o & 0_No <= x
  proof
    assume x in L_NonNegativePart o;
    then ex l be Surreal st x= l & l in L_o & 0_No <= l by Def1;
    hence thesis;
  end;
  x in R_NonNegativePart o implies x in R_o & 0_No <= x
  proof
    assume x in R_NonNegativePart o;
    then ex l be Surreal st x= l & l in R_o & 0_No <= l by Def1;
    hence thesis;
  end;
  hence thesis by A1,Def1;
end;
