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;
reserve r,r1,r2 for Real;

theorem Th61:
  x==y & x is *real implies y is *real
proof
   assume
A1:x==y & x is *real;
   then consider n be Nat such that
A2: uInt.-n < x < uInt.n;
A3: uInt.-n < y < uInt.n by A2,A1,SURREALO:4;
   x == real_qua x == real_qua y by A1,Lm20;
   then x== real_qua y by SURREALO:4;
   then y== real_qua y by A1,SURREALO:4;
   hence thesis by A3;
end;
