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<b>let </b><font color="Maroon">S</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_9.html#K2">Subspaces_of</a> <font color="Maroon">V</font></span>)</span>, <font color="Maroon">T</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="vectsp_9.html#K2">Subspaces_of</a> <font color="Maroon">V</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:12_1_2"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff  ex <font color="Olive">W1</font>, <font color="Olive">W2</font> being   <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Olive">W2</font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Olive">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="vectsp_9.html#K1">dim</a> <font color="Olive">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">X</font> <a href="hidden.html#R1">=</a>  <a href="pencil_4.html#K3" target="_self">pencil</a> <font color="Olive">W1</font>,<font color="Olive">W2</font>,<font color="Maroon">k</font> ) )
 <b>and </b><br/><a NAME="E2:12_1_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> iff  ex <font color="Olive">W1</font>, <font color="Olive">W2</font> being   <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Olive">W2</font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Olive">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="vectsp_9.html#K1">dim</a> <font color="Olive">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Olive">X</font> <a href="hidden.html#R1">=</a>  <a href="pencil_4.html#K3" target="_self">pencil</a> <font color="Olive">W1</font>,<font color="Olive">W2</font>,<font color="Maroon">k</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:12_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:12_1_2_1_1"/><b>then </b>
 ex <font color="Olive">W1</font>, <font color="Olive">W2</font> being   <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Olive">W2</font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Olive">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="vectsp_9.html#K1">dim</a> <font color="Olive">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pencil_4.html#K3" target="_self">pencil</a> <font color="Olive">W1</font>,<font color="Olive">W2</font>,<font color="Maroon">k</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:12_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:12_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/><a NAME="E3:12_1_2_1"/><b>then </b>
 ex <font color="Olive">W1</font>, <font color="Olive">W2</font> being   <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Olive">W2</font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1">dim</a> <font color="Olive">W1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="vectsp_9.html#K1">dim</a> <font color="Olive">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pencil_4.html#K3" target="_self">pencil</a> <font color="Olive">W1</font>,<font color="Olive">W2</font>,<font color="Maroon">k</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:12_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:12_1_2"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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