reserve k, l, m, n, i, j for Nat,
  K, N for non empty Subset of NAT,
  Ke, Ne, Me for Subset of NAT,
  X,Y for set;
reserve f for Function of Segm n,Segm k;
reserve x,y for set;

theorem Th51:
  n choose 1=n & n choose 2=n*(n-1)/2 & n choose 3=n*(n-1)*(n-2)/6
  & n choose 4=n*(n-1)*(n-2)*(n-3)/24
proof
  now
    (n = 0 or ... or n = 3) or 3 < n;
    then per cases;
    suppose
      n=0;
      hence thesis by NEWTON:def 3;
    end;
    suppose
      n=1;
      hence thesis by NEWTON:21,def 3;
    end;
    suppose
      n=2;
      hence thesis by NEWTON:21,23,def 3;
    end;
    suppose
A1:   n=3;
      then reconsider n1=n-1,n2=n-2 as Element of NAT by NAT_1:20;
A2:   (n2+1)!=(n2)!*(n2+1) by NEWTON:15;
A3:   (n2!)/(n2!)=1 by XCMPLX_1:60;
      (n1+1)!=(n1)!*n by NEWTON:15;
      then n choose 2=((n2)!*((n-1)*n))/((n2!)*(2!)) by A1,A2,NEWTON:def 3
        .=((n2!)/(n2!)*((n-1)*n))/2 by NEWTON:14,XCMPLX_1:83
        .=((n-1)*n)/2 by A3;
      hence thesis by A1,NEWTON:21,23,def 3;
    end;
    suppose
A4:   n>3;
      then n>=3+1 by NAT_1:13;
      then reconsider n1=n-1,n2=n-2,n3=n-3,n4=n-4 as Element of NAT by NAT_1:21
,XXREAL_0:2;
A5:   (n1+1)!=(n1)!*n by NEWTON:15;
A6:   (n2+1)!=(n2)!*(n2+1) by NEWTON:15;
A7:   (n2!)/(n2!)=1 by XCMPLX_1:60;
      n>=2 by A4,XXREAL_0:2;
      then
A8:   n choose 2=((n2)!*(n1*n))/((n2!)*(2!)) by A5,A6,NEWTON:def 3
        .=((n2!)/(n2!)*(n1*n))/2 by NEWTON:14,XCMPLX_1:83
        .=n*n1/2 by A7;
A9:   (n4!)/(n4!)=1 by XCMPLX_1:60;
A10:  (n4+1)!=(n4)!*(n4+1) by NEWTON:15;
A11:  (n3!)/(n3!)=1 by XCMPLX_1:60;
A12:  (n3+1)!=(n3)!*(n3+1) by NEWTON:15;
      then
A13:  n choose 3=((n3)!*(n2*n1*n))/((n3!)*(3!)) by A4,A5,A6,NEWTON:def 3
        .=((n3!)/(n3!)*(n2*n1*n))/6 by Th50,XCMPLX_1:83
        .=n*n1*n2/6 by A11;
      n>=3+1 by A4,NAT_1:13;
      then
n choose 4=((n4!)*(n3*n2*n1*n))/((n4!)*(4!)) by A5,A6,A12,A10,NEWTON:def 3
        .=((n4!)/(n4!)*(n3*n2*n1*n))/24 by Th50,XCMPLX_1:83
        .=n*n1*n2*n3/24 by A9;
      hence thesis by A4,A8,A13,NEWTON:23,XXREAL_0:2;
    end;
  end;
  hence thesis;
end;
