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

theorem Th73:
  for s be Surreal st r <> 0 holds
    x is (s,y,r)_term iff
    |. x - (s + uReal.r* No_omega^ y).| infinitely< No_omega^y
proof
  let s be Surreal such that
A1: r <>0;
  set N =No_omega^ y, R = uReal.r,sNR=s+R*N;
  set X = x - sNR;
  sNR-sNR==0_No by SURREALR:39;
  then
A2: sNR +X = x+(sNR+-sNR)==x+0_No = x by SURREALR:37,43;
  thus x is (s,y,r)_term implies
  |. X.| infinitely< N
  proof
    assume x is (s,y,r)_term;
    then sNR +X is (s,y,r)_term by A2,Th72;
    hence |.X.| infinitely< N by Th71;
  end;
  assume |.X.| infinitely< N;
  then sNR +X is (s,y,r)_term by A1,Th71;
  hence thesis by A2,Th72;
end;
