 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 Th33:
  not x == 0_No implies x * x" == 1_No
proof
  assume
A1:not x == 0_No;
  per cases;
  suppose 0_No < x;
    then
A2: x is positive;
    then x" = inv x by A1,Def14;
    hence thesis by A2,Lm2;
  end;
  suppose
A3: not 0_No < x;
    then
A4: -x is positive by A1,SURREALR:23,10;
    then reconsider I=inv (-x) as Surreal;
A5: not x is positive by A3;
    (x")*x = (-(x")) * (-x) by SURREALR:58
    .= I *(-x) by A5,A1,Def14;
    hence thesis by A4,Lm2;
  end;
end;
