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 Th33:
  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 Th8;
      seq.n<>0 & seq1.n<>0 by A1,Th5;
      hence (seq(#)seq1).n<>0 by A2,XCMPLX_1:6;
    end;
    hence thesis by Th5;
  end;
  assume
A3: seq(#)seq1 is non-zero;
  now
    let n;
    (seq(#)seq1).n=(seq.n)*(seq1.n) by Th8;
    hence seq.n<>0 by A3,Th5;
  end;
  hence seq is non-zero by Th5;
  now
    let n;
    (seq(#)seq1).n=(seq.n)*(seq1.n) by Th8;
    hence seq1.n<>0 by A3,Th5;
  end;
  hence thesis by Th5;
end;
