 reserve A,B,O for Ordinal,
      n,m for Nat,
      a,b,o for object,
      x,y,z for Surreal,
      X,Y,Z for set,
      Inv,I1,I2 for Function;

theorem Th18:
  x is positive implies x == ||.x.||
proof
  assume
A1: x is positive;
  then consider y be Surreal such that
A2:y==x and
A3: (for z holds z in L_y iff z=0_No or (z in L_x & 0_No<z) ) and
A4: (for z holds z in R_y iff z in R_x & 0_No<z ) by Lm1;
  0_No < y by A1,A2,SURREALO:4;
  then reconsider y as positive Surreal by Def8;
  (z in L_y iff z=0_No or (z in L_x & z is positive)) &
  (z in R_y iff z in R_x & z is positive) by A3,A4;
  hence thesis by A2,A1,Def9;
end;
