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 Th50:
  l>0 implies ex x st ((a,b) In_Power (k+l)).(k+1) = a*x
  proof
    assume l>0;
    then consider n such that
    A3: l = 1+n by NAT_1:10,14;
    consider x such that
    A4: x = ((k+l) choose k)*(a|^n)*(b|^k);
    ((a,b) In_Power (k+l)).(k+1)=((k+l) choose k)*(a|^l)*(b|^k) by Lm1
    .= ((k+l) choose k)*(a|^1*a|^n)*(b|^k) by A3,NEWTON:8
    .= a*x by A4;
    hence thesis;
  end;
