reserve A,B for Ordinal,
        o for object,
        x,y,z for Surreal,
        n for Nat,
        r,r1,r2 for Real;

theorem Th39:
  not x == 0_No implies |. -x .| = |.x.|
proof
  assume not x == 0_No;
  then per cases;
  suppose
A1: 0_No < x;
    then
A2: |. -x .| = - - x by SURREALR:10,23,Def6;
    0_No <= x by A1;
    hence thesis by Def6,A2;
  end;
  suppose A3: x < 0_No;
    then 0_No <= -x by SURREALR:10,23;
    then |. -x .| = -x by Def6;
    hence thesis by A3,Def6;
  end;
end;
