theorem Th23:
  t <= u & u <= t implies u = t
proof
  set R = the Relation of A;
  R well_orders QC-symbols(A) \ NAT by Def32;
  then
A1: R is_antisymmetric_in QC-symbols(A) \ NAT by WELLORD1:def 5;
  assume
A2: t <= u & u <= t;
  per cases;
    suppose
A3:   t in NAT & u in NAT;
      then consider n,m such that
A4:    t = n & u = m & n <= m by A2,Def33;
      consider k,j such that
A5:    u = k & t = j & k <= j by A2,A3,Def33;
      thus thesis by A4,A5,XXREAL_0:1;
    end;
    suppose not t in NAT or not u in NAT;
      then per cases;
        suppose not t in NAT;
          then
A6:        not t in NAT & not u in NAT by A2,Def33;
          then
A7:        t in QC-symbols(A) \ NAT & u in QC-symbols(A) \ NAT
            by XBOOLE_0:def 5;
          [t,u] in R & [u,t] in R by A2,A6,Def33;
          hence u = t by A1,A7,RELAT_2:def 4;
        end;
        suppose not u in NAT;
          then
A8:        not t in NAT & not u in NAT by A2,Def33;
          then
A9:        t in QC-symbols(A) \ NAT & u in QC-symbols(A) \ NAT
            by XBOOLE_0:def 5;
          [t,u] in R & [u,t] in R by A2,A8,Def33;
          hence u = t by A1,A9,RELAT_2:def 4;
        end;
    end;
end;
