reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;
reserve n,k for Nat;
reserve M,N for non empty TopSpace;

theorem Th13:
  for x1,x2 being FinSequence of REAL,i being Element of NAT
  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 REAL,
  i be Element of NAT;
  assume
A1: i in dom mlt(x1,x2);
A2: mlt(x1,x2)= multreal.:(x1,x2) by RVSUM_1:def 9;
  dom multreal=[:REAL, REAL:] & rng x1 c= REAL by FUNCT_2:def 1;
  then [:rng x1, rng x2:] c= dom multreal by ZFMISC_1:96;
  then
A3: 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
A4: x2/.i=x2.i by PARTFUN1:def 6;
  i in dom x1 by A1,A3,XBOOLE_0:def 4;
  then x1/.i=x1.i by PARTFUN1:def 6;
  hence mlt(x1,x2).i = (x1/.i)*(x2/.i) by A4,RVSUM_1:59;
  hence thesis by A1,PARTFUN1:def 6;
end;
