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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>() <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c2">V</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b1">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b2">W2</font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b1">W1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_4.html#K3" title="PENCIL_4:func.3">pencil</a> <font color="Olive" title="b1">W1</font>,<font color="Olive" title="b2">W2</font>,<font color="Maroon" title="c3">k</font> );<br/>
<b>consider </b><font color="Maroon" title="c4">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:12_1_1"/><i><font color="Green" title="E12">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:12_1_1"/>
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/>

<b>hence </b><a NAME="E2:12_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="pencil_4.html#E2:12_1_1"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">X</font> = <font color="Maroon" title="c4">X</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c2">V</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c5">X</font>
;<br/>

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E5:12_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> implies  ex <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> st <br/>( <font color="Olive" title="b1">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b2">W2</font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b1">W1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Olive" title="b2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_4.html#K3" title="PENCIL_4:func.3">pencil</a> <font color="Olive" title="b1">W1</font>,<font color="Olive" title="b2">W2</font>,<font color="Maroon" title="c3">k</font> ) )
 <b>by </b><i><a class="txt" href="pencil_4.html#E2:12_1_1"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>

<b>given </b><font color="Maroon" title="c7">W1</font>, <font color="Maroon" title="c8">W2</font> being    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><b> such that </b><a NAME="E6:12_1_1"/><i><font color="Green" title="E13">A2</font></i>: 
( <font color="Maroon" title="c7">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c8">W2</font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c7">W1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c8">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_4.html#K3" title="PENCIL_4:func.3">pencil</a> <font color="Maroon" title="c7">W1</font>,<font color="Maroon" title="c8">W2</font>,<font color="Maroon" title="c3">k</font> )
 ;<br/>

<b>thus </b><a NAME="E7:12_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E6:12_1_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="pencil_4.html#E2:12_1_1"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>


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