reserve l, m, n for Nat;
reserve a,b for Int-Location,
  f for FinSeq-Location,
  s,s1,s2 for State of SCM+FSA;
reserve L for finite Subset of Int-Locations;
reserve L for finite Subset of FinSeq-Locations;
reserve L for finite Subset of Int-Locations;

theorem Th24:
 for n,m being Element of NAT holds
  n < m implies min ((RWNotIn-seq L).n) < min ((RWNotIn-seq L).m)
proof let n,m be Element of NAT;
  set RL = RWNotIn-seq L;
  now
    let n be Element of NAT;
    defpred X[Nat] means
 n < $1 implies min (RL.n) < min (RL.$1);
A1: for m being Nat st X[m] holds X[m+1]
    proof
      let m be Nat such that
A2:   n < m implies min (RL.n) < min (RL.m);
      assume n < m+1;
      then
A3:   n <= m by NAT_1:13;
      per cases by A3,XXREAL_0:1;
      suppose
        n = m;
        hence min (RL.n) < min (RL.(m+1)) by Th23;
      end;
      suppose
        n < m;
        hence min (RL.n) < min (RL.(m+1)) by A2,Th23,XXREAL_0:2;
      end;
    end;
A4: X[0];
    thus for n being Nat holds X[n] from NAT_1:sch 2(A4, A1);
  end;
  hence thesis;
end;
