reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;
reserve z for Complex;
reserve x,y for Complex;
reserve r,s,t for natural Number;

theorem Th25:
  s choose r is Element of NAT
proof
A0: s is Nat by TARSKI:1;
  defpred P[Nat] means for r holds ($1 choose r) is Element of NAT;
A1: for s be Nat st P[s] holds P[s+1]
  proof
    let s be Nat;
    assume
A2: P[s];
A3: for r st r <> 0 holds ((s+1) choose r) is Element of NAT
    proof
      let r;
      assume
A4:   r <> 0;
      ((s+1) choose r) is Element of NAT
      proof
        consider t being Nat such that
A5:     r = t+1 by A4,NAT_1:6;
        reconsider t as Element of NAT by ORDINAL1:def 12;
        reconsider m1 = (s choose t), m2 = (s choose (t+1)) as Element of NAT
        by A2;
        m1 + m2 =(s choose t) + (s choose (t+1));
        hence thesis by A5,Th22;
      end;
      hence thesis;
    end;
    let r;
    r = 0 or r<>0;
    hence thesis by A3,Th19;
  end;
A6: P[0]
  proof
    let r;
    r = 0 or r > 0;
    hence thesis by Def3,Th19;
  end;
  for s be Nat holds P[s] from NAT_1:sch 2(A6,A1);
  hence thesis by A0;
end;
