
theorem Th24:
  for n, m, k being Element of NAT holds PFCrt (n,m) c= PFArt (k,m ) iff n < k
proof
  let n, m, k be Element of NAT;
  thus PFCrt (n,m) c= PFArt (k,m) implies n < k
  proof
    assume
A1: PFCrt (n,m) c= PFArt (k,m);
    assume
A2: k <= n;
A3: not [2*n+1,m] in PFArt (k,m)
    proof
      assume
A4:   [2*n+1,m] in PFArt (k,m);
      per cases by A4,Def2;
      suppose
A5:     ex m9 being odd Element of NAT st m9 <= 2*k & [m9,m] = [2*n+1, m];
A6:     2*k <= 2*n by A2,NAT_1:4;
        2*n+1 <= 2*k by A5,XTUPLE_0:1;
        hence thesis by A6,NAT_1:13;
      end;
      suppose
        [2*k,m] = [2*n+1,m];
        hence thesis by XTUPLE_0:1;
      end;
    end;
    [2*n+1,m] in PFCrt (n,m) by Def3;
    hence thesis by A1,A3;
  end;
  thus thesis by Lm7;
end;
