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 Th9:
  seq1 = r(#)seq2 iff for n holds seq1.n=r*seq2.n
proof
  thus seq1 = r(#)seq2 implies for n holds seq1.n=r*seq2.n by VALUED_1:6;
  assume for n holds seq1.n=r*seq2.n;
  then
A1: for n being object st n in dom seq1 holds seq1.n = r * seq2.n;
  dom seq1 = NAT by FUNCT_2:def 1
    .= dom seq2 by FUNCT_2:def 1;
  hence thesis by A1,VALUED_1:def 5;
end;
