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 Th37:
  r<>0c & seq is non-zero implies r(#)seq is non-zero
proof
  assume that
A1: r<>0c and
A2: seq is non-zero and
A3: not r(#)seq is non-zero;
  consider n1 such that
A4: (r(#)seq).n1=0c by A3,Th4;
A5: r*seq.n1=0c by A4,VALUED_1:6;
  seq.n1 <> 0c by A2,Th4;
  hence contradiction by A1,A5;
end;
