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

theorem Th69:
  for s be Surreal st r <>0 holds
       s + uReal.r* No_omega^ y is (s,y,r)_term
proof
  let s be Surreal such that
A1:r <> 0;
  set R=uReal.r,N=No_omega^ y;
  set sRNs = (s + R* N) + - s;
  s - s == 0_No by SURREALR:39;
  then
A2: sRNs = R* N + (s +- s) == R* N + 0_No =R* N
  by SURREALR:43,SURREALR:37;
A3: not R* N == 0_No by A1,Th67;
  then
A4: not sRNs ==0_No by A2,SURREALO:4;
A5: omega-y (R* N) = Unique_No y by A1,Th68;
  |.R*N.| is positive by A1,Th67,Th36;
  then |. sRNs .|, |.R*N.| are_commensurate by A2,Th48,Th8;
  then
A6: omega-y (sRNs ) = Unique_No y ==y
  by SURREALO:def 10,A4,A5,Th61,A3;
  then No_omega^ omega-y sRNs == N by Lm5;
  then (No_omega^ omega-y sRNs) *R== N*R by SURREALR:51;
  then - (No_omega^ omega-y sRNs) *R== - N*R by SURREALR:10;
  then sRNs - (No_omega^ omega-y sRNs) * R == N*R - N*R ==0_No
  by A2,SURREALR:43,SURREALR:39;
  then sRNs +- (No_omega^ omega-y sRNs) * R ==0_No by SURREALO:4;
  then |.sRNs +- (No_omega^ omega-y sRNs) * R.| == 0_No by Def6;
  then |. sRNs - (No_omega^ omega-y sRNs) * R .|
  infinitely< |.sRNs.| by A4, Th64;
  hence thesis by A1,A4,Def8,A6;
end;
