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 Th4:
  for x1,x2 being FinSequence of G,i st i in dom mlt(x1,x2) holds
  mlt(x1,x2).i = (x1/.i)*(x2/.i) & (mlt(x1,x2))/.i=(x1/.i)*(x2/.i)
proof
  let x1,x2 be FinSequence of G,i;
  assume
A1: i in dom mlt(x1,x2);
A2: mlt(x1,x2)=(the multF of G).:(x1,x2) by FVSUM_1:def 7;
A3: rng x2 c= the carrier of G by FINSEQ_1:def 4;
  dom (the multF of G)=[:the carrier of G,the carrier of G:] & rng x1 c=
  the carrier of G by FINSEQ_1:def 4,FUNCT_2:def 1;
  then [:rng x1, rng x2:] c= dom the multF of G by A3,ZFMISC_1:96;
  then
A4: dom mlt(x1,x2) = dom x1 /\ dom x2 by A2,FUNCOP_1:69;
  then i in dom x2 by A1,XBOOLE_0:def 4;
  then
A5: x2/.i=x2.i by PARTFUN1:def 6;
  i in dom x1 by A1,A4,XBOOLE_0:def 4;
  then x1/.i=x1.i by PARTFUN1:def 6;
  hence mlt(x1,x2).i = (x1/.i)*(x2/.i) by A1,A5,FVSUM_1:60;
  hence thesis by A1,PARTFUN1:def 6;
end;
