reserve

  k,n,m,i,j for Element of NAT,
  K for Field;
reserve L for non empty addLoopStr;
reserve G for non empty multLoopStr;

theorem
  for f1,f2 being FinSequence of K holds dom mlt(f1,f2) = dom f1 /\ dom
  f2 & for i st i in dom (mlt(f1,f2)) holds (mlt(f1,f2)).i = (f1/.i) * (f2/.i)
proof
  let f1,f2 be FinSequence of K;
A1: rng (f2) c= the carrier of K by FINSEQ_1:def 4;
  dom (the multF of K)=[:the carrier of K,the carrier of K:] & rng (f1) c=
  the carrier of K by FINSEQ_1:def 4,FUNCT_2:def 1;
  then mlt(f1,f2)=(the multF of K).:(f1,f2) & [:rng (f1), rng (f2):] c= dom (
  the multF of K) by A1,FVSUM_1:def 7,ZFMISC_1:96;
  hence dom (mlt(f1,f2))= dom (f1) /\ dom (f2) by FUNCOP_1:69;
  thus thesis by Th4;
end;
