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<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:26_1_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c3">A</font> is  <a href="vectsp_7.html#V2" title="VECTSP_7:attr.2">base</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="vectsp_7.html#D4" onmouseover="rs('vectsp_7/D4')" onmouseout="rh()">VECTSP_7:def 4</a></span>;<br/>
<span class="kw">reconsider </span><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"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c4">A</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="vectsp_7.html#D3" onmouseover="rs('vectsp_7/D3')" onmouseout="rh()">VECTSP_7:def 3</a></span>;<br/>
<a NAME="E5:26_1_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="lmod_7.html#T4" target="_self" onmouseover="rs('lmod_7/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font> as   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">reconsider </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><font color="Maroon" title="c9">H</font> =  <a href="vectsp_7.html#K1" title="VECTSP_7: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="E8:26_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/>
<a NAME="E9:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T45" onmouseover="rs('xboole_1/T45')" onmouseout="rh()">XBOOLE_1:45</a></span>;<br/>
<a NAME="E10:26_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" onmouseover="rs('xboole_1/T79')" onmouseout="rh()">XBOOLE_1:79</a></span>;<br/>
<a NAME="E11:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<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="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c7">A1</font>, <a href="vectsp_7.html#K1" title="VECTSP_7: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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E9:26_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="lmod_7.html#T6" target="_self" onmouseover="rs('lmod_7/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E12:26_1_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
 ex <font color="Olive" title="b1">a</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">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="struct_0.html#K4" title="STRUCT_0:func.4">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="vectsp_7.html#K1" title="VECTSP_7: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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c6">a</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">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="vectsp_7.html#K1" title="VECTSP_7: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> )</span><br/>

<span class="kw">thus </span><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="struct_0.html#K4" title="STRUCT_0:func.4">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="vectsp_7.html#K1" title="VECTSP_7: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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:26_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E11:26_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="lmod_7.html#T1" target="_self" onmouseover="rs('lmod_7/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="lmod_6.html#T6" onmouseover="rs('lmod_6/T6')" onmouseout="rh()">LMOD_6:6</a>, <a class="ref" href="lmod_6.html#D4" onmouseover="rs('lmod_6/D4')" onmouseout="rh()">LMOD_6:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">take </span>
 <a href="vectsp_7.html#K1" title="VECTSP_7: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>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">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="struct_0.html#K4" title="STRUCT_0:func.4">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="vectsp_7.html#K1" title="VECTSP_7: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> )</span><br/>

<span class="kw">thus </span><a NAME="E13:26_1_1"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">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="struct_0.html#K4" title="STRUCT_0:func.4">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="vectsp_7.html#K1" title="VECTSP_7: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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
