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;

theorem Th5:
  (a*x+b*y) .|. z = a * x .|. z + b * y .|. z
proof
  (a*x+b*y) .|. z = (a*x) .|. z + (b*y) .|. z by Def2
    .= a * x .|. z + (b*y) .|. z by Def2;
  hence thesis by Def2;
end;
