reserve i1 for Element of INT;
reserve j1,j2,j3 for Integer;
reserve p,s,k,n for Nat;
reserve x,y,xp,yp for set;
reserve G for Group;
reserve a,b for Element of G;
reserve F for FinSequence of G;
reserve I for FinSequence of INT;

theorem Th4:
  for I being FinSequence of INT holds Product(((len I)|->a)|^I) = a|^Sum I
proof
  defpred P[FinSequence of INT] means Product(((len $1) |->a)|^($1))=a|^Sum($1
  );
A1: for p being FinSequence of INT for x being Element of INT st P[p] holds
  P[p^<*x*>] by Lm3;
A2: P[<*> INT] by Lm2;
  for p being FinSequence of INT holds P[p] from FINSEQ_2:sch 2(A2,A1);
  hence thesis;
end;
