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<b>let </b><font color="Maroon">S</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="vectsp_5.html#K3">Subspaces</a> <font color="Maroon">V</font></span>)</span>, <font color="Maroon">T</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="vectsp_5.html#K3">Subspaces</a> <font color="Maroon">V</font></span>)</span>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_2_1"/>
<font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font>
 ;<br/><b>assume </b><b>that </b><br/><a NAME="E2:7_1_2_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">W</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> holds <br/> ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff ( <font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Olive">W</font> &amp; <font color="Olive">W</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font> ) )
 <b>and </b><br/><a NAME="E3:7_1_2_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">W</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> holds <br/> ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> iff ( <font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Olive">W</font> &amp; <font color="Olive">W</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font> ) )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_1_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="E1:7_1_2_1_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="7_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_2_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:7_1_2_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3">VECTSP_5:def 3</a></i>;<br/>
<a NAME="E4:7_1_2_1_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff ( <font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:7_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:7_1_2_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:7_1_2_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3">VECTSP_5:def 3</a></i>;<br/><a NAME="E5:7_1_2_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> iff ( <font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:7_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:7_1_2_1"/>
<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/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:7_1_2"/>
( ( <font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font> &amp; ( for <font color="Olive">W</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> holds <br/> ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff ( <font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Olive">W</font> &amp; <font color="Olive">W</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font> ) ) ) &amp; ( for <font color="Olive">W</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> holds <br/> ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> iff ( <font color="Maroon">W1</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Olive">W</font> &amp; <font color="Olive">W</font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font> ) ) ) implies <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> ) &amp; ( <font color="Maroon">W1</font> is  not    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W2</font> &amp; <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> ) )
 ;<br/>


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