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 Th32:
  seq is non-zero implies seq" is non-zero
proof
  assume that
A1: seq is non-zero and
A2: not seq" is non-zero;
  consider n1 such that
A3: (seq").n1=0 by A2,Th5;
  (seq.n1)"=(seq").n1 by VALUED_1:10;
  hence contradiction by A1,A3,Th5,XCMPLX_1:202;
end;
