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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">W1</font> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">W2</font> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v2</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">u1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">u2</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>















<b>assume </b><a NAME="E1:61"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">V</font> <a href="rlsub_2.html#R1" target="_self">is_the_direct_sum_of</a> <font color="Maroon">W1</font>,<font color="Maroon">W2</font>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:61"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u2</font> )
 <b>and </b><br/><a NAME="E3:61"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">v1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">u1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W1</font> )
 <b>and </b><br/><a NAME="E4:61"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">v2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">u2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">C2</font> = <font color="Maroon">v1</font> <a href="rlsub_1.html#K3">+</a> <font color="Maroon">W2</font> as    <a href="rlsub_1.html#M2">Coset</a> of <font color="Maroon">W2</font> <b>by </b><i><a class="ref" href="rlsub_1.html#D6">RLSUB_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W1</font> as    <a href="rlsub_1.html#M2">Coset</a> of <font color="Maroon">W1</font> <b>by </b><i><a class="ref" href="rlsub_1.html#T90">RLSUB_1:90</a></i>;<br/>
<i><font color="Green">E38</font></i>: <font color="Maroon">u1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font>
<b>by </b><i>, <a class="txt" href="rlsub_2.html#E4:61"><i><font color="Green">E37</font></i></a></i>
<br/>.= 
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E8:61"/>
<font color="Maroon">v2</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">u2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">W2</font>
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E12"><i><font color="Green">Lemma52</font></i></a>, <a class="ref" href="rlsub_1.html#T31">RLSUB_1:31</a></i>;<br/>
<a NAME="E9:61"/><b>then </b>
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> )
 
<b>by </b><i><a class="txt" href="rlsub_2.html#E11"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="rlsub_2.html#T10" target="_self">Th10</a>, <a class="ref" href="rlsub_1.html#T59">RLSUB_1:59</a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:61"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">u</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E12:61"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="rlsub_2.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E13:61"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> )
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:61"/>
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font>
 ;<br/>

<b>thus </b><a NAME="E15:61"/>
<font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font>
 <b>by </b><i>, <a class="txt" href="rlsub_2.html#E12:61"><i><font color="Green">E56</font></i></a>, <a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>;<br/>


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