reserve X for non empty CUNITSTR;
reserve a, b for Complex;
reserve x, y for Point of X;
reserve X for ComplexUnitarySpace;
reserve x, y, z, u, v for Point of X;

theorem Th37:
  ||.x.|| = 0 iff x = 0.X
proof
  thus ||.x.|| = 0 implies x = 09(X)
  proof
    0 <= Re (x.|.x) by Def11;
    then
A1: 0 <= |.(x.|.x).| by Th29;
    assume ||.x.|| = 0;
    then |.(x.|.x).| = 0 by A1,SQUARE_1:24;
    then x.|.x = 0c by COMPLEX1:45;
    hence thesis by Def11;
  end;
  assume x = 09(X);
  hence thesis by Def11,COMPLEX1:44,SQUARE_1:17;
end;
