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<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>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="parsp_1.html#K6" title="PARSP_1:func.6">4C3</a> <font color="Maroon" title="c1">F</font> &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">a</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">c</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">d</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">c</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">d</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">a</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">a</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">b</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">c</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">d</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">c</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">d</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">a</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">a</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">b</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">c</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">d</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">c</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">d</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">a</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">b</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font> ) );<br/>
<b>let </b><font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c3">H2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:29_1_2"/><i><font color="Green" title="E21">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="c2">H1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 <b>and </b><br/><a NAME="E2:29_1_2"/><i><font color="Green" title="E22">A2</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="c3">H2</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 ;<br/>

<a NAME="E3:29_1_2"/>
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="c2">H1</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">H2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:29_1_2_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>] )
 
<b>by </b><i><a class="txt" href="parsp_1.html#E1:29_1_2"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:29_1_2_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H1</font> iff <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">H2</font> )
 <b>by </b><i><a class="txt" href="parsp_1.html#E2:29_1_2"><i><font color="Green" title="E22">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:29_1_2"/>
<font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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