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
  seq is non-zero & seq9 is non-zero implies seq1/"seq + seq19/"seq9=(
seq1(#)seq9+seq19(#)seq)/"(seq(#)seq9) & seq1/"seq - seq19/"seq9=(seq1(#)seq9-
  seq19(#)seq)/"(seq(#)seq9)
proof
  assume that
A1: seq is non-zero and
A2: seq9 is non-zero;
  thus seq1/"seq + seq19/"seq9=(seq1(#)seq9)/"(seq(#)seq9)+seq19/"seq9 by A2
,Th36
    .=(seq1(#)seq9)/"(seq(#)seq9)+(seq19(#)seq)/"(seq(#)seq9) by A1,Th36
    .=(seq1(#)seq9+seq19(#)seq)(#)((seq(#)seq9)") by Th9
    .=(seq1(#)seq9+seq19(#)seq)/"(seq(#)seq9);
  thus seq1/"seq - seq19/"seq9=(seq1(#)seq9)/"(seq(#)seq9)-seq19/"seq9 by A2
,Th36
    .=(seq1(#)seq9)/"(seq(#)seq9)-(seq19(#)seq)/"(seq(#)seq9) by A1,Th36
    .=(seq1(#)seq9-seq19(#)seq)(#)((seq(#)seq9)") by Th14
    .=(seq1(#)seq9-seq19(#)seq)/"(seq(#)seq9);
end;
