reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

theorem Th54:
  for n be prime Nat, a,b be positive Nat holds
    n*a*b divides (((a,b) In_Power n)|n/^1).k
  proof
    let n be prime Nat, a,b be positive Nat;
L1: not k in dom (((a,b) In_Power n)|n/^1) implies
      n*a*b divides ((((a,b)In_Power n)|n)/^1).k
    proof
      assume not k in dom (((a,b) In_Power n)|n/^1); then
      ((((a,b) In_Power n)|n)/^1).k = {} by FUNCT_1:def 2;
      hence thesis by NAT_D:6;
    end;
    n is prime & k in dom (((a,b) In_Power n)|n/^1) implies
      n*a*b divides ((((a,b) In_Power n)|n)/^1).k
    proof
      A0: k = (k+1) - 1;
      assume
      A1: n is prime & k in dom (((a,b) In_Power n)|n/^1); then
      A2:(((a,b) In_Power n)|n/^1) is not empty;
      A3: k >= 1 & k <= len (((a,b) In_Power n)|n/^1) by FINSEQ_3:25,A1; then
      A3a: k < n by A2,Th20,XXREAL_0:2; then
      A4: k+1 < n+1 & k+1 > 1 by A1,FINSEQ_3:25,NAT_1:13,XREAL_1:6;
      A4a: n-k > k-k by A3a,XREAL_1:9;
      consider l such that
      A4b: n = k + l by A3a,NAT_1:10;
      A4d: l = n-k by A4b;
      len ((a,b) In_Power n) = n+1 by NEWTON:def 4; then
      A6: k+1 in dom ((a,b) In_Power n) by A4, FINSEQ_3:25;
      A7: ((((a,b) In_Power n)|n)/^1).(k) = ((a,b) In_Power n).(k+1)
        by A1,Lm5
      .= (n choose k) * a|^l * b|^k by A0,A4d,A6,NEWTON:def 4;
      A8: n divides (n choose k) by A3,A3a,Th21;
      a divides a|^l & b divides b|^k by A4a,A4b,A3,NAT_3:3; then
      a*b divides a|^l*b|^k by NAT_3:1; then
      n*(a*b) divides (n choose k) *(a|^l * b|^k) by A8,NAT_3:1;
      hence thesis by A7;
    end;
    hence thesis by L1;
  end;
