Let $X$ a vector space on $\mathbb{R}$ and denote with $\theta$ the zero vector, i must prove that $$0x=\theta$$ for all $x\in X$  Now,  $0x=(0+0)x=0x+0x\Rightarrow$ $0x=\theta$  Correct?
Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well.  The following proof is solely based on vector space related axioms. Axiom names are italicised. They are defined in Wikipedia (see vector space article).  Vector spaces - Multiplying by zero scalar yields zero vector  \begin{array}{lrll} \text{Let} &amp; \dots &amp; \text{be} &amp; \dots \\ \hline &amp; F &amp;&amp; \text{a field.} \\ &amp; V &amp;&amp; \text{a vector space over $F$.} \\ &amp; 0 &amp;&amp; \text{an identity element of addition of $F$.} \\ &amp; \mathbf{0} &amp;&amp; \text{an identity element of addition of $V$.} \\ &amp; \mathbf{v} &amp;&amp; \text{an arbitrary vector in $V$.} \\ \end{array}  $$\text{Then, }0\mathbf{v} = \mathbf{0}.$$  Proof. We will denote by $1$ an identity element of scalar multiplication; we will denote by $(-\mathbf{v})$ an additive inverse of $\mathbf{v}$. \begin{align*} 0\mathbf{v} &amp;= 0\mathbf{v} + \mathbf{0}                     &amp;&amp; \text{by }\textit{Identity element of vector addition} \\ &amp;= 0\mathbf{v} + (\mathbf{v} + (-\mathbf{v}))   &amp;&amp; \text{by }\textit{Inverse elements of vector addition} \\ &amp;= (0\mathbf{v} + \mathbf{v}) + (-\mathbf{v})   &amp;&amp; \text{by }\textit{Associativity of vector addition} \\ &amp;= (0\mathbf{v} + 1\mathbf{v}) + (-\mathbf{v})  &amp;&amp; \text{by }\textit{Identity element of scalar multiplication} \\ &amp;= ((0 + 1)\mathbf{v}) + (-\mathbf{v})          &amp;&amp; \text{by }\textit{Distributivity of scalar multiplication (field addition)} \\ &amp;= ((1 + 0)\mathbf{v}) + (-\mathbf{v})          &amp;&amp; \text{by }\textit{Commutativity of field addition} \\ &amp;= (1\mathbf{v}) + (-\mathbf{v})                &amp;&amp; \text{by }\textit{Identity element of field addition} \\ &amp;= \mathbf{v} + (-\mathbf{v})                   &amp;&amp; \text{by }\textit{Identity element of scalar multiplication} \\ &amp;= \mathbf{0}                                   &amp;&amp; \text{by }\textit{Inverse elements of vector addition} \\ \end{align*} QED
Changing planes at Newark Airport I fly in from Sweden with United to Newark Liberty Airport. The connecting flight with United departs to Costa Rica a few hours later from Newark as well. The two flight-segments are part of the same trip and ticket.  I was hoping to stay in the same terminal but am suspecting that it wont be as straightforward as I had hoped - reading through the older posts it seems that a transfer is needed from A to C but it's not fully clear.   Do I need to pass through immigration? Do I need to fetch our bags? Even if my bags have tags all the way to SJO? Do I need to change terminals? Which terminal would I arrive in? Depart from? Returning multiple values from a C++ function Is there a difference between single and double quotes in Java? Is there a difference between single and double quotes in Java? Can an Indian travel to India with 50 days before passport expires? 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I've seen the following links however, the I'm not sure about relavancy of the soluions given i.e.,          Regarding my visualforce component, I'm doing just one DML operation on a custom setting from a callout response. Also, I'm not passing any attribute to the component.   To give a background of the component, its just a list view of records from a different system retrieved through a callout on load via JavaScript.  Let me know if in case anything else is needed.   Thanks in Advance! Peace! Does the new finding on "reversing a quantum jump mid-flight" rule out any interpretations of QM? new finding by Minev et al. seems to suggest that transitions between atomic states are not instantaneous, but continuous processes wherein a superposition smoothly adjusts from favoring one state to another (if I understand it correctly). The authors also claim to be able to catch a system "mid-jump" and reverse it. Popular articles are  and .  I am curious if this finding rules out any interpretations of QM. It seems to generally go against the Copenhagen attitude, which describes measurements as collapsing physical systems into a definite classical state. The popular articles indeed claim that the founders of QM would have been surprised by the new finding.  The link with the asterisk mentions that something called "quantum trajectories theory" predicts what was observed. Is this an interpretation, or a theory? And are they implying that other interpretations/theories don't work? Combining two sorted lists in Python I have two lists of objects. Each list is already sorted by a property of the object that is of the datetime type. I would like to combine the two lists into one sorted list. Is the best way just to do a sort or is there a smarter way to do this in Python? Can I sell starting gear in Adventurers League play? I'm starting a new AL game (as a player) and I have a question that's not addressed in the AL material.  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However, we've still got the forward implication, because the converse of an injection is always a partial surjection.  A specific question. Define that two sets are weakly equipotent iff $X \succsim Y$ and $Y \succsim X$. Question: does weak equipotence imply genuine equipotence? I'm guessing "no."  A general question: In broad terms, how would the structure of the cardinal numbers in ZF be altered by adopting $\succsim$ rather than $\lesssim$ as the fundamental order relation, and taking equality of cardinal numbers to correspond to weak equipotence of sets? I'm guessing this would hide a lot of the (usually very complicated) structure of the cardinal numbers in ZF.  Remark. The order symbols can be written \lesssim and \succsim.