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 Th42:
  r<>0 & seq is non-zero implies r(#)seq is non-zero
proof
  assume that
A1: r<>0 and
A2: seq is non-zero and
A3: not r(#)seq is non-zero;
  consider n1 such that
A4: (r(#)seq).n1=0 by A3,Th5;
A5: seq.n1 <> 0 by A2,Th5;
  r*seq.n1=0 by A4,Th9;
  hence contradiction by A1,A5,XCMPLX_1:6;
end;
