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 Th41:
  seq1 is non-zero implies seq2/"seq=(seq2(#)seq1)/"(seq(#)seq1)
proof
  assume
A1: seq1 is non-zero;
  now
    let n be Element of NAT;
A2: seq1.n<>0 by A1,Th5;
    thus (seq2/"seq).n=(seq2.n)*1*seq".n by Th8
      .=(seq2.n)*((seq1.n)*(seq1.n)")*seq".n by A2,XCMPLX_0:def 7
      .=(seq2.n)*(seq1.n)*((seq1.n)"*seq".n)
      .=((seq2(#)seq1).n)*((seq1.n)"*seq".n) by Th8
      .=((seq2(#)seq1).n)*((seq1".n)*seq".n) by VALUED_1:10
      .=((seq2(#)seq1).n)*(seq"(#)seq1").n by Th8
      .=((seq2(#)seq1).n)*(seq(#)seq1)".n by Th34
      .=((seq2(#)seq1)/"(seq(#)seq1)).n by Th8;
  end;
  hence thesis by FUNCT_2:63;
end;
