
theorem
  for n, k being Element of NAT, x being set st x in PFBrt (n+1,k) holds
  ex y being set st y in PFBrt (n,k) & y c= x
proof
  let n, k be Element of NAT, x be set;
  assume
A1: x in PFBrt (n+1,k);
  per cases by A1,Def4;
  suppose
    ex m being non zero Element of NAT st m <= n+1 & x = PFArt (m,k);
    then consider m being non zero Element of NAT such that
A2: m <= n+1 and
A3: x = PFArt (m,k);
    thus ex y being set st y in PFBrt (n,k) & y c= x
    proof
      per cases by A2,NAT_1:8;
      suppose
A4:     m <= n;
        take y = x;
        thus y in PFBrt (n,k) by A3,A4,Def4;
        thus thesis;
      end;
      suppose
A5:     m = n+1;
        take PFCrt (n,k);
        n < n+1 by NAT_1:13;
        hence thesis by A3,A5,Def4,Lm7;
      end;
    end;
  end;
  suppose
A6: x = PFCrt (n+1,k);
    take y = PFCrt (n,k);
    thus y in PFBrt (n,k) by Def4;
    thus thesis by A6,Lm6;
  end;
end;
