reserve x, x1, x2, y, X, D for set,
  i, j, k, l, m, n, N for Nat,
  p, q for XFinSequence of NAT,
  q9 for XFinSequence,
  pd, qd for XFinSequence of D;

theorem Th29:
  2*m <= n implies card Domin_0(n,m) = (n+1-2*m) / (n+1-m) * (n choose m)
proof
  assume
A1: 2*m<=n;
  now
    per cases;
    suppose
A2:   m=0;
      then (n choose m)=1 by NEWTON:19;
      then (n+1-2*m)/(n+1-m)*(n choose m)=1 by A2,XCMPLX_1:60;
      hence thesis by A2,Th24;
    end;
    suppose
A3:   m>0;
A4:   m<=m+m by NAT_1:11;
      then reconsider nm=n-m as Nat by A1,NAT_1:21,XXREAL_0:2;
      reconsider m1=m-1 as Nat by A3,NAT_1:20;
      set n9=n!;
      set m9=m!;
      set nm19=(nm+1)!;
      set nm9=nm!;
      m<=n by A1,A4,XXREAL_0:2;
      then
A5:   n choose m=n9/(m9*nm9) by NEWTON:def 3;
A6:   2*(m1+1)<=n by A1;
      set m19=m1!;
A7:   1/(m19*nm19)=((m1+1)*1)/((m19*nm19)*(m1+1)) by XCMPLX_1:91
        .=m/(nm19*(m19*(m1+1)))
        .=m/(nm19*((m1+1)!)) by NEWTON:15
        .=-(-m)/(nm19*m9) by XCMPLX_1:190;
      1/(m9*nm9)=((nm+1)*1)/((m9*nm9)*(nm+1)) by XCMPLX_1:91
        .=(nm+1)/(m9*(nm9*(nm+1)))
        .=(nm+1)/(m9*nm19) by NEWTON:15;
      then
A8:   1/(m9*nm9)-1/(m19*nm19)=(nm+1)/(m9*nm19)+(-m)/(m9*nm19) by A7
        .=((nm+1)+(-m))/(m9*nm19) by XCMPLX_1:62
        .=(n+1-2*m)/(m9*(nm9*(nm+1))) by NEWTON:15
        .=(1*(n+1-2*m))/((m9*nm9)*(nm+1))
        .=(1/(m9*nm9))*((n+1-2*m)/(nm+1)) by XCMPLX_1:76;
      m1<=m1+(1+m1+1) by NAT_1:11;
      then
A9:   m1 <= n by A1,XXREAL_0:2;
      n-m1=nm+1;
      then
A10:  n choose m1=n9/(m19*nm19) by A9,NEWTON:def 3;
      n9/(m9*nm9)-n9/(m19*nm19)=n9*(1/(m9*nm9))-n9/(m19*nm19) by XCMPLX_1:99
        .=n9*(1/(m9*nm9))-n9*(1/(m19*nm19)) by XCMPLX_1:99
        .=n9*(1/(m9*nm9)-1/(m19*nm19))
        .=n9*((1/(m9*nm9))*((n+1-2*m)/(nm+1))) by A8
        .=(n9*(1/(m9*nm9)))*((n+1-2*m)/(nm+1))
        .=((n9*1)/(m9*nm9))*((n+1-2*m)/(nm+1)) by XCMPLX_1:74;
      hence thesis by A5,A10,A6,Th28;
    end;
  end;
  hence thesis;
end;
