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<b>consider </b><font color="Maroon" title="c3">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E2:26_1_1"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c3">A</font> is <a href="mod_3.html#V1" title="MOD_3:attr.1">base</a>
 <b>by </b><i><a class="txt" href="lmod_7.html#E2:26"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="mod_3.html#D3" title="MOD_3:def.3">MOD_3:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">A</font> = <font color="Maroon" title="c3">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> ;<br/>
<a NAME="E4:26_1_1"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c4">A</font> is <a href="lmod_5.html#V1" title="LMOD_5:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="txt" href="lmod_7.html#E2:26_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="mod_3.html#D2" title="MOD_3:def.2">MOD_3:def 2</a></i>;<br/>
<a NAME="E5:26_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="lmod_7.html#E1:26"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="lmod_7.html#E2:26_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="lmod_7.html#T4" target="_self" title="LMOD_7:th.4">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font>;<br/>
<a NAME="E7:26_1_1"/><i><font color="Green" title="E14">A6</font></i>: 
not <font color="Maroon" title="c1">K</font> is <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a>
 
<b>by </b><i><a class="txt" href="lmod_7.html#E1:26"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="lmod_6.html#T6" title="LMOD_6:th.6">LMOD_6:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c5">x</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="lmod_7.html#E5:26_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">A1</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> ;<br/>
<b>set </b><font color="Maroon" title="c8">A2</font> = <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">A1</font>;<br/>
<b>set </b><font color="Maroon" title="c9">H</font> =  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">A1</font></span>)</span>;<br/>
<a NAME="E10:26_1_1"/>
<font color="Maroon" title="c7">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="txt" href="lmod_7.html#E5:26_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E11:26_1_1"/><b>then </b><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">A1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E12:26_1_1"/>
<font color="Maroon" title="c7">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">A1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E13:26_1_1"/><b>then </b><i><font color="Green" title="E16">A8</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a>  <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <font color="Maroon" title="c7">A1</font>, <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">A1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lmod_7.html#E2:26_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="lmod_7.html#E11:26_1_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="lmod_7.html#T6" target="_self" title="LMOD_7:th.6">Th6</a></i>;<br/>
<a NAME="E14:26_1_1"/><i><font color="Green" title="E17">A9</font></i>: 
 ex <font color="Olive" title="b1">a</font> being  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c2">V</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <span class="p1"><a href="lmod_6.html#K3" title="LMOD_6:func.3">&lt;:</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="lmod_6.html#K3" title="LMOD_6:func.3">:&gt;</a></span>, <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">A1</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" title="c6">a</font>
;<br/>

<b>thus </b><a NAME="E1:26_1_1_1"/>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c2">V</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <span class="p1"><a href="lmod_6.html#K3" title="LMOD_6:func.3">&lt;:</a><span class="default"><font color="Maroon" title="c6">a</font></span><a href="lmod_6.html#K3" title="LMOD_6:func.3">:&gt;</a></span>, <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">A1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="lmod_7.html#E4:26_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="lmod_7.html#E5:26_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="lmod_7.html#E7:26_1_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="lmod_7.html#E13:26_1_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="lmod_7.html#T1" target="_self" title="LMOD_7:th.1">Th1</a>, <a class="ref" href="lmod_6.html#D6" title="LMOD_6:def.6">LMOD_6:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
 <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">A1</font></span>)</span>
;<br/>

<b>thus </b><a NAME="E15:26_1_1"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c2">V</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <span class="p1"><a href="lmod_6.html#K3" title="LMOD_6:func.3">&lt;:</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="lmod_6.html#K3" title="LMOD_6:func.3">:&gt;</a></span>, <a href="mod_3.html#K1" title="MOD_3:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">A1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="lmod_7.html#E14:26_1_1"><i><font color="Green" title="E17">A9</font></i></a></i>;<br/>


</div>
