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 Th75:
  No_uOrdinal_op A < No_uOrdinal_op B iff A in B
proof
A1: No_uOrdinal_op A == No_Ordinal_op A &
  No_uOrdinal_op B == No_Ordinal_op B by Th73;
  thus No_uOrdinal_op A < No_uOrdinal_op B implies A in B
  proof
    assume No_uOrdinal_op A < No_uOrdinal_op B;
    then No_Ordinal_op A < No_uOrdinal_op B by A1,SURREALO:4;
    then No_Ordinal_op A < No_Ordinal_op B by A1,SURREALO:4;
    hence thesis by Th68;
  end;
  assume A in B;
  then No_uOrdinal_op A < No_Ordinal_op B by Th68,A1,SURREALO:4;
  hence thesis by A1,SURREALO:4;
end;
