reserve f for Function;
reserve n,k,n1 for Nat;
reserve r,p for Real;
reserve x,y,z for object;
reserve seq,seq1,seq2,seq3,seq9,seq19 for Real_Sequence;

theorem Th8:
  seq = seq1 (#) seq2 iff for n holds seq.n =seq1.n * seq2.n
proof
  thus seq = seq1 (#) seq2 implies for n holds seq.n =seq1.n * seq2.n
  proof
    assume
A1: seq = seq1 (#) seq2;
    let n;
A2: n in NAT by ORDINAL1:def 12;
    dom seq = NAT by FUNCT_2:def 1;
    hence thesis by A1,VALUED_1:def 4,A2;
  end;
  assume for n holds seq.n =seq1.n * seq2.n;
  then
A3: for n being object st n in dom seq holds seq.n = seq1.n * seq2.n;
  dom seq = NAT /\ NAT by FUNCT_2:def 1
    .= NAT /\ dom seq2 by FUNCT_2:def 1
    .= dom seq1 /\ dom seq2 by FUNCT_2:def 1;
  hence thesis by A3,VALUED_1:def 4;
end;
