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 Th27:
  ||.a * x.|| = |.a.| * ||.x.||
proof
A1: 0 <= a^2 & 0 <= x .|. x by Def2,XREAL_1:63;
  ||.a * x.|| = sqrt (a * (x .|. (a * x))) by Def2
    .= sqrt (a * (a * (x .|. x))) by Def2
    .= sqrt (a^2 * (x .|. x))
    .= sqrt (a^2) * sqrt (x .|. x) by A1,SQUARE_1:29
    .= |.a.| * sqrt (x .|. x) by COMPLEX1:72;
  hence thesis;
end;
