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<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:7_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>() <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="vectsp_5.html#K3">Subspaces</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being   <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>3</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>4</sub></font> );<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">H<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:7_1_1_1"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="subset_1.html#S1">SUBSET_1:sch 1</a>();<br/></i><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>thus </b><a NAME="E4:7_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> implies ( <font color="Maroon">c<sub>3</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>4</sub></font> ) )
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:7_1_1_1"><i><font color="Green">E7</font></i></a></i>;<br/><b>assume </b><a NAME="E5:7_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E6:7_1_1_1"/><b>then </b><i><font color="Green">E8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>]
 ;<br/><a NAME="E7:7_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp_5.html#K3">Subspaces</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3">VECTSP_5:def 3</a></i>;<br/><b>hence </b><a NAME="E8:7_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:7_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pencil_4.html#E6:7_1_1_1"><i><font color="Green">E8</font></i></a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:7_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> is  not    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as   <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">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><b>thus </b><a NAME="E3:7_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div>
<b>end;</b></div>


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