reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;
reserve d,d1,d2 for Dyadic;
reserve i,j for Integer,
        n,m,p for Nat;

theorem Th37:
  x is uSurreal & born x is finite iff ex d be Dyadic st x = uDyadic.d
proof
  hereby
    assume
A1: x is uSurreal & born x is finite;
    then reconsider b = born x as Nat;
    x in Day b by SURREAL0:def 18;
    then ex d be Dyadic st
    x == uDyadic.d & uDyadic.d in Day b by Th35;
    hence ex d be Dyadic st x = uDyadic.d by A1,SURREALO:50;
  end;
  given d be Dyadic such that
A2:x = uDyadic.d;
  consider n such that
A3:uDyadic.d in Day n by Th36;
  born uDyadic.d c= n by A3,SURREAL0:def 18;
  hence thesis by A2;
end;
