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 Th67:
  No_Ordinal_op A in Day A
proof
  defpred P[Ordinal] means No_Ordinal_op $1 in Day $1;
A1: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
  proof
    let D be Ordinal;
    assume
A2: for C be Ordinal st C in D holds P[C];
    per cases;
    suppose D is limit_ordinal;
      then consider X be set such that
A3:   No_Ordinal_op D = [X,{}] &
      (for o holds o in X iff ex B st B in D & o = No_Ordinal_op B) by Th66;
A4:   X<<{};
      for o be object st o in X\/ {} ex O st O in D & o in Day O
      proof
        let o;
        assume o in X \/ {};
        then consider B such that
A5:     B in D & o = No_Ordinal_op B by A3;
        No_Ordinal_op B in Day B by A5,A2;
        hence thesis by A5;
      end;
      hence thesis by A3,A4,SURREAL0:46;
    end;
    suppose not D is limit_ordinal;
      then consider B such that
A6:   succ B = D by ORDINAL1:29;
A7:   No_Ordinal_op B in Day B by A6,ORDINAL1:6,A2;
      then reconsider OB=No_Ordinal_op B as Surreal;
A8:   {OB}<< {};
      for o be object st o in {OB} \/ {} ex O st O in D & o in Day O
      proof
        let o;
        assume o in {OB} \/ {};
        then o=OB by TARSKI:def 1;
        hence thesis by A7,A6,ORDINAL1:6;
      end;
      then [{OB},{}] in Day D by A8,SURREAL0:46;
      hence thesis by A6,Th65;
    end;
  end;
  for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
  hence thesis;
end;
