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 Th26:
  ||.x.|| = 0 iff x = 0.X
proof
  thus ||.x.|| = 0 implies x = 09(X)
  proof
    assume
A1: ||.x.|| = 0;
    0 <= x .|. x by Def2;
    then x .|. x = 0 by A1,SQUARE_1:24;
    hence thesis by Def2;
  end;
  assume x = 09(X);
  hence thesis by Def2,SQUARE_1:17;
end;
