reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;

theorem Th19:
  seq.(n + 1) = Sum(seq, n + 1) - Sum(seq, n)
proof
  thus Sum(seq, n + 1) - Sum(seq, n) = ( seq.(n + 1) + Sum(seq, n) ) - Sum(seq
  , n) by Def1
    .= seq.(n + 1) + ( Sum(seq, n) - Sum(seq, n) ) by RLVECT_1:def 3
    .= seq.(n + 1) + 0.X by RLVECT_1:15
    .= seq.(n + 1);
end;
