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
  seq is non-zero implies (seq1/"seq)(#)seq=seq1
proof
  assume
A1: seq is non-zero;
  now
    let n be Element of NAT;
A2: seq.n<>0 by A1,Th5;
    thus ((seq1/"seq)(#)seq).n=((seq1(#)seq").n)*seq.n by Th8
      .=(seq1.n)*(seq".n)*seq.n by Th8
      .=(seq1.n)*(seq.n)"*seq.n by VALUED_1:10
      .=(seq1.n)*((seq.n)"*seq.n)
      .=(seq1.n)*1 by A2,XCMPLX_0:def 7
      .=seq1.n;
  end;
  hence thesis by FUNCT_2:63;
end;
