reserve A for QC-alphabet;
reserve k,n,m for Nat;
reserve P for QC-pred_symbol of A;
reserve F for Element of QC-WFF(A);
reserve Q for QC-pred_symbol of A;
reserve F, G for (Element of QC-WFF(A)), s for FinSequence;
reserve p for Element of QC-WFF(A);
reserve F for Element of QC-WFF(A);
reserve p for Element of QC-WFF(A);
reserve j,k for Nat;
reserve k for Nat;
reserve s,t,u,v for QC-symbol of A;

theorem Th24:
  t <= u or u <= t
proof
  set R = the Relation of A;
  R well_orders QC-symbols(A) \ NAT by Def32;
  then R is_connected_in QC-symbols(A) \ NAT & R is_reflexive_in
   QC-symbols(A) \ NAT by WELLORD1:def 5;
  then
A1: R is_strongly_connected_in QC-symbols(A) \ NAT by ORDERS_1:7;
  per cases;
    suppose
A2:   t in NAT & u in NAT;
      then consider n,m such that
A3:    n = t & m = u;
      n <= m or m <= n;
      hence thesis by A3,Def33,A2;
    end;
    suppose not t in NAT or not u in NAT;
      then per cases;
        suppose
A4:        not t in NAT;
           per cases;
           suppose u in NAT;
             hence thesis by A4,Def33;
           end;
           suppose
A5:          not u in NAT;
             then t in QC-symbols(A) \NAT & u in QC-symbols(A) \NAT
              by A4,XBOOLE_0:def 5;
             then [t,u] in R or [u,t] in R by A1,RELAT_2:def 7;
             hence thesis by A4,A5,Def33;
           end;
        end;
        suppose
A6:        not u in NAT;
           per cases;
           suppose t in NAT;
             hence thesis by A6,Def33;
           end;
           suppose
A7:          not t in NAT;
             then t in QC-symbols(A) \NAT & u in QC-symbols(A) \NAT
              by A6,XBOOLE_0:def 5;
             then [u,t] in R or [t,u] in R by A1,RELAT_2:def 7;
             hence thesis by A6,A7,Def33;
           end;
        end;
    end;
end;
