reserve N for Nat;
reserve n,m,n1,n2 for Nat;
reserve q,r,r1,r2 for Real;
reserve x,y for set;
reserve w,w1,w2,g,g1,g2 for Point of TOP-REAL N;
reserve seq,seq1,seq2,seq3,seq9 for Real_Sequence of N;

theorem Th21:
  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.TOP-REAL N by A3,Th3;
A5: seq.n1 <> 0.TOP-REAL N by A2,Th3;
  r*seq.n1=0.TOP-REAL N by A4,Th5;
  hence contradiction by A1,A5,RLVECT_1:11;
end;
