reserve X for non empty UNITSTR;
reserve a, b for Real;
reserve x, y for Point of X;
reserve X for RealUnitarySpace;
reserve x, y, z, u, v for Point of X;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve  n for Nat;

theorem
  seq1 is constant implies a * seq1 is constant
proof
  assume
A1: seq1 is constant;
    set seq = a * seq1;
  consider x such that
A2: for n being Nat holds seq1.n = x by A1;
  take z = a * x;
  let n be Nat;
  thus seq.n = a * seq1.n by NORMSP_1:def 5
    .= z by A2;
end;
