reserve f for Function;
reserve n,k,n1 for Element of NAT;
reserve r,p for Complex;
reserve x,y for set;
reserve seq,seq1,seq2,seq3,seq9,seq19 for Complex_Sequence;

theorem Th28:
  seq is non-zero & seq1 is non-zero iff seq(#)seq1 is non-zero
proof
  thus seq is non-zero & seq1 is non-zero implies seq(#)seq1 is non-zero
  proof
    assume
A1: seq is non-zero & seq1 is non-zero;
    now
      let n;
A2:   (seq(#)seq1).n=(seq.n)*(seq1.n) by VALUED_1:5;
      seq.n<>0c & seq1.n<>0c by A1,Th4;
      hence (seq(#)seq1).n<>0c by A2;
    end;
    hence thesis by Th4;
  end;
  assume
A3: seq(#)seq1 is non-zero;
  now
    let n;
    (seq(#)seq1).n=(seq.n)*(seq1.n) by VALUED_1:5;
    hence seq.n<>0c by A3,Th4;
  end;
  hence seq is non-zero by Th4;
  now
    let n;
    (seq(#)seq1).n=(seq.n)*(seq1.n) by VALUED_1:5;
    hence seq1.n<>0c by A3,Th4;
  end;
  hence thesis by Th4;
end;
