reserve f for Function;
reserve n,k,n1 for Element of NAT;
reserve r,p for Complex;
reserve x,y for set;
reserve seq,seq1,seq2,seq3,seq9,seq19 for Complex_Sequence;

theorem Th36:
  seq1 is non-zero implies seq2/"seq=(seq2(#)seq1)/"(seq(#)seq1)
proof
  assume
A1: seq1 is non-zero;
  now
    let n;
A2: seq1.n<>0c by A1,Th4;
    thus (seq2/"seq).n=(seq2.n)*1r*seq".n by VALUED_1:5
      .=(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 VALUED_1:5
      .=((seq2(#)seq1).n)*((seq1".n)*seq".n) by VALUED_1:10
      .=((seq2(#)seq1).n)*(seq"(#)seq1").n by VALUED_1:5
      .=((seq2(#)seq1).n)*(seq(#)seq1)".n by Th29
      .=((seq2(#)seq1)/"(seq(#)seq1)).n by VALUED_1:5;
  end;
  hence thesis by FUNCT_2:63;
end;
