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 Th12:
  seq1 = abs seq iff for n holds seq1.n= |.seq.n.|
proof
  thus seq1 = abs seq implies for n holds seq1.n=|.seq.n.| by VALUED_1:18;
  assume for n holds seq1.n= |.seq.n.|;
  then
A1: for n being object st n in dom seq1 holds seq1.n = |.seq.n.|;
  dom seq1 = NAT by FUNCT_2:def 1
    .= dom seq by FUNCT_2:def 1;
  hence thesis by A1,VALUED_1:def 11;
end;
