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
  seq1/"seq + seq19/"seq = (seq1 + seq19) /" seq & seq1/"seq - seq19/"
  seq = (seq1 - seq19) /" seq
proof
  thus seq1/"seq + seq19/"seq = (seq1 + seq19) (#) (seq") by Th15
    .= (seq1 + seq19) /" seq;
  thus seq1/"seq - seq19/"seq = (seq1 - seq19) (#) (seq") by Th20
    .= (seq1 - seq19) /" seq;
end;
