reserve n for Nat,
        i,j,i1,i2,i3,i4,i5,i6 for Element of n,
        p,q,r for n-element XFinSequence of NAT;
reserve i,j,n,n1,n2,m,k,l,u,e,p,t for Nat,
        a,b for non trivial Nat,
        x,y for Integer,
        r,q for Real;

theorem Th9:
   i <= j & 2*j <= n+1 implies n choose i <= n choose j
proof
  defpred P[Nat] means
  i <= $1 & 2*$1 <= n+1 implies n choose i <= n choose $1;
A1: P[0];
A2: P[k] implies P[k+1]
  proof
    set k1=k+1;
    assume
A3:   P[k] & i <= k1 & 2*k1 <= n+1;
    then per cases by XXREAL_0:1;
    suppose i=k1;
      hence thesis;
    end;
    suppose A4: i < k1;
      k < k1 by NAT_1:13;
      then
A5:     2*k <= 2*k1 by XREAL_1:66;
      k1+k1 <= n+1 by A3;
      then k1 <= n+1-k1 by XREAL_1:19;
      then (n-k)/k1 >=1 by XREAL_1:181;
      then ((n-k)/k1)*(n choose k) >= 1*(n choose k) by XREAL_1:66;
      then n choose k1 >= n choose k by IRRAT_1:5;
      hence thesis by A5,A4,NAT_1:13,A3,XXREAL_0:2;
    end;
  end;
  P[k] from NAT_1:sch 2(A1,A2);
  hence thesis;
end;
