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
  a * Partial_Sums(seq1) + b * Partial_Sums(seq2) = Partial_Sums(a *
  seq1 + b * seq2)
proof
  thus a * Partial_Sums(seq1) + b * Partial_Sums(seq2) = Partial_Sums(a * seq1
  ) + b * Partial_Sums(seq2) by Th3
    .= Partial_Sums(a * seq1) + Partial_Sums(b * seq2) by Th3
    .= Partial_Sums(a * seq1 + b * seq2) by Th1;
end;
