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

theorem Th71:
  for s be Surreal st r <>0 holds
       s + uReal.r* No_omega^ y+x is (s,y,r)_term iff
         |.x.| infinitely< No_omega^ y
proof
  let s be Surreal such that A1:r <>0;
  set N = No_omega^ y,R=uReal.r,sRNx = s + R* N +x;
  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;
  not R* N == 0_No by A1,Th67;
  then
A3: not sRNs ==0_No by A2,SURREALO:4;
A4: |. sRNs.| is positive by A3,Th36;
A5: |.sRNs.|,|.N*R.| are_commensurate by A2,Th48,A4,Th8;
A6: |.N*R.|,N are_commensurate by A1,Th66;
  then
A7: |.sRNs.|,N are_commensurate by Th4,A5;
A8: sRNx +-s = s + (R* N +x) +-s by SURREALR:37
  .= s+-s + (R* N +x) by SURREALR:37;
  s-s ==0_No by SURREALR:39;
  then
A9: s+-s + (R* N +x) == 0_No + (R* N +x) =R* N +x by SURREALR:43;
  sRNs + x == R* N +x by A2,SURREALR:43;
  then
A10: sRNx +- s == sRNs +x by A9,SURREALO:4,A8;
A11: s + R* N is (s,y,r)_term by A1,Th69;
  then
A12: not sRNs == 0_No & omega-y (sRNs) == y & omega-r (sRNs) = r;
  thus sRNx is (s,y,r)_term implies |.x.| infinitely< N
  proof
    assume
A13: sRNx is (s,y,r)_term;
    then
A14: not sRNx +- s == 0_No &
     omega-y (sRNx +- s) == y & omega-r (sRNx + - s) = r;
A15: not sRNs+x ==0_No by A10,A14,SURREALO:4;
A16: omega-y (sRNx +- s) = omega-y (sRNs +x) &
    omega-r (sRNx + - s) = omega-r (sRNs + x) by A13,A10,Th70;
A17: omega-r (sRNs + x) = omega-r (sRNs) by A13,A11,A10,Th70;
    omega-y (sRNs)==omega-y (sRNs+x) by A14,A12,A16,SURREALO:4;
    then omega-y sRNs = omega-y (sRNs + x) by SURREALO:50;
    hence thesis by A7,Th16,A17,A15,A11,Th62;
  end;
  assume |.x.| infinitely< N;
  then |.x.| infinitely< |.sRNs.| by A6,Th4,A5,Th16;
  then
A18: not sRNs+x == 0_No &
  omega-y sRNs = omega-y (sRNs + x) &
  omega-r sRNs = omega-r (sRNs+x) by Th63;
  then not sRNx +- s == 0_No by A10,SURREALO:4;
  hence thesis by A18,A10,Th70,A11;
end;
