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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 is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ;<br/>
<a NAME="E1:7_1_1"/><i><font color="Green" title="E4">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/>



<a NAME="E1:7_1_1_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="E1:7_1_1_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/>

<a NAME="E2:7_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:7_1_1_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:7_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>)] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>)] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:7_1_1"/><i><font color="Green" title="E5">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">a</font>)]
 
;<br/>
<a NAME="E3:7_1_1"/><i><font color="Green" title="E6">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">b</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:7_1_1_2"/><i><font color="Green" title="E7">A4</font></i>: 
( <font color="Maroon" title="c2">a</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  &amp; <font color="Maroon" title="c3">b</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  )
 ;<br/>

<a NAME="E2:7_1_1_2"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">c</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="E1:7_1_1_2_1"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:7_1_1_2_1"/><i><font color="Green" title="E8">A5</font></i>: 
( <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:7_1_1_2_1"/>
<font color="Maroon" title="c5">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> 
 
<b>by </b><i><a class="txt" href="zf_model.html#E1:7_1_1_2"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="zf_model.html#E3:7_1_1_2_1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:7_1_1_2_1"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> 
 <b>by </b><i><a class="txt" href="zf_model.html#E3:7_1_1_2_1"><i><font color="Green" title="E8">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:7_1_1"/><i><font color="Green" title="E7">A6</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">x</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">a</font>)]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/>



<b>assume </b><a NAME="E1:7_1_1_3"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c2">a</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> 
 ;<br/>

<a NAME="E2:7_1_1_3"/>
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">b</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="E1:7_1_1_3_1"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:7_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c5">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:7_1_1_3_1"/><i><font color="Green" title="E9">A8</font></i>: 
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:7_1_1_3_1"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="zf_model.html#E4:7_1_1_3_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:7_1_1_3_1"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> 
 <b>by </b><i><a class="txt" href="zf_model.html#E1:7_1_1_3"><i><font color="Green" title="E8">A7</font></i></a>, <a class="txt" href="zf_model.html#E4:7_1_1_3_1"><i><font color="Green" title="E9">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">a</font>)]
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:7_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>6</sub></font>(<font color="Maroon" title="c1">H</font>)]
 <b>from </b><i><a class="ref" href="zf_model.html#S4" target="_self" title="ZF_MODEL:sch.4">ZF_MODEL:sch 4</a>(<a class="txt" href="zf_model.html#E2"><i><font color="Green" title="E2">Lm1</font></i></a>, <a class="txt" href="zf_model.html#E1:7_1_1"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="zf_model.html#E2:7_1_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="zf_model.html#E3:7_1_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="zf_model.html#E4:7_1_1"><i><font color="Green" title="E7">A6</font></i></a>);</i><br/>


</div>
