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<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:26_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="mod_3.html#V1">base</a>
 <b>by </b><i><a class="txt" href="lmod_7.html#E2:26"><i><font color="Green">E10</font></i></a>, <a class="ref" href="mod_3.html#D3">MOD_3:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E4:26_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="lmod_5.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="txt" href="lmod_7.html#E2:26_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="mod_3.html#D2">MOD_3:def 2</a></i>;<br/>
<a NAME="E5:26_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="lmod_7.html#E1:26"><i><font color="Green">E9</font></i></a>, <a class="txt" href="lmod_7.html#E2:26_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="lmod_7.html#T4" target="_self">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E7:26_1_1"/><i><font color="Green">E14</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="realset2.html#V3">trivial</a>
 
<b>by </b><i><a class="txt" href="lmod_7.html#E1:26"><i><font color="Green">E9</font></i></a>, <a class="ref" href="lmod_6.html#T6">LMOD_6:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="lmod_7.html#E5:26_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="mod_3.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>;<br/>
<a NAME="E10:26_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="lmod_7.html#E5:26_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E11:26_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E12:26_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E13:26_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="vectsp_5.html#R1">is_the_direct_sum_of</a>  <a href="mod_3.html#K1">Lin</a> <font color="Maroon">c<sub>7</sub></font>, <a href="mod_3.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lmod_7.html#E2:26_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="lmod_7.html#E11:26_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="lmod_7.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E14:26_1_1"/><i><font color="Green">E17</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="vectsp_5.html#R1">is_the_direct_sum_of</a> <span class="p1"><a href="lmod_6.html#K3">&lt;:</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lmod_6.html#K3">:&gt;</a></span>, <a href="mod_3.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E1:26_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="vectsp_5.html#R1">is_the_direct_sum_of</a> <span class="p1"><a href="lmod_6.html#K3">&lt;:</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="lmod_6.html#K3">:&gt;</a></span>, <a href="mod_3.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="lmod_7.html#E4:26_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="lmod_7.html#E5:26_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="lmod_7.html#E7:26_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="lmod_7.html#E13:26_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="lmod_7.html#T1" target="_self">Th1</a>, <a class="ref" href="lmod_6.html#D6">LMOD_6:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
 <a href="mod_3.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
;<br/>

<b>thus </b><a NAME="E15:26_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="vectsp_5.html#R1">is_the_direct_sum_of</a> <span class="p1"><a href="lmod_6.html#K3">&lt;:</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="lmod_6.html#K3">:&gt;</a></span>, <a href="mod_3.html#K1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="lmod_7.html#E14:26_1_1"><i><font color="Green">E17</font></i></a></i>;<br/>


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