reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;

theorem Th12:
  for i be Integer holds - uInt.i = uInt.-i
proof
  defpred P[Nat] means - uInt.$1 = uInt.-$1;
  uInt.0=0_No by Def1;
  then
A1: P[0];
A2: P[n] implies P[n+1]
  proof
    assume
A3: P[n];
    uInt.(n+1) = [{uInt.n},{}] by Def1;
    then L_(uInt.(n+1)) = {uInt.n} & R_(uInt.(n+1)) = {};
    then - uInt.(n+1) = [{},-- {uInt.n}] by SURREALR:22,7
    .= [{},{uInt.-n}] by A3,SURREALR:21
    .= uInt.-(n+1) by Def1;
    hence thesis;
  end;
A4: P[n] from NAT_1:sch 2(A1,A2);
  let i be Integer;
  i in INT by INT_1:def 2;
  then consider o be Nat such that
A5:i=o or i=-o by INT_1:def 1;
  P[o] by A4;
  hence thesis by A5;
end;
