<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">Z</font>, <font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">X2</font>, <font color="Maroon" title="c4">X3</font>, <font color="Maroon" title="c5">X4</font>, <font color="Maroon" title="c6">X5</font>, <font color="Maroon" title="c7">X6</font>, <font color="Maroon" title="c8">X7</font>, <font color="Maroon" title="c9">X8</font>, <font color="Maroon" title="c10">X9</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



















<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E20">A1</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> iff  ex <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">x3</font>, <font color="Olive" title="b5">x4</font>, <font color="Olive" title="b6">x5</font>, <font color="Olive" title="b7">x6</font>, <font color="Olive" title="b8">x7</font>, <font color="Olive" title="b9">x8</font>, <font color="Olive" title="b10">x9</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X1</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font> &amp; <font color="Olive" title="b4">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X3</font> &amp; <font color="Olive" title="b5">x4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X4</font> &amp; <font color="Olive" title="b6">x5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X5</font> &amp; <font color="Olive" title="b7">x6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X6</font> &amp; <font color="Olive" title="b8">x7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X7</font> &amp; <font color="Olive" title="b9">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X8</font> &amp; <font color="Olive" title="b10">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X9</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_6.html#K1" title="MCART_6:func.1">[</a><span class="default"><font color="Olive" title="b2">x1</font>,<font color="Olive" title="b3">x2</font>,<font color="Olive" title="b4">x3</font>,<font color="Olive" title="b5">x4</font>,<font color="Olive" title="b6">x5</font>,<font color="Olive" title="b7">x6</font>,<font color="Olive" title="b8">x7</font>,<font color="Olive" title="b9">x8</font>,<font color="Olive" title="b10">x9</font></span><a href="mcart_6.html#K1" title="MCART_6:func.1">]</a></span> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:41_1"/>
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> implies <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="mcart_5.html#K2" title="MCART_5:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">X3</font>,<font color="Maroon" title="c5">X4</font>,<font color="Maroon" title="c6">X5</font>,<font color="Maroon" title="c7">X6</font>,<font color="Maroon" title="c8">X7</font>,<font color="Maroon" title="c9">X8</font></span><a href="mcart_5.html#K2" title="MCART_5:func.2">:]</a></span>,<font color="Maroon" title="c10">X9</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c13">x2</font>, <font color="Maroon" title="c14">x3</font>, <font color="Maroon" title="c15">x4</font>, <font color="Maroon" title="c16">x5</font>, <font color="Maroon" title="c17">x6</font>, <font color="Maroon" title="c18">x7</font>, <font color="Maroon" title="c19">x8</font>, <font color="Maroon" title="c20">x9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:41_1_1"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X1</font> &amp; <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font> &amp; <font color="Maroon" title="c14">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X3</font> &amp; <font color="Maroon" title="c15">x4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X4</font> &amp; <font color="Maroon" title="c16">x5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X5</font> &amp; <font color="Maroon" title="c17">x6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X6</font> &amp; <font color="Maroon" title="c18">x7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X7</font> &amp; <font color="Maroon" title="c19">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X8</font> &amp; <font color="Maroon" title="c20">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X9</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_6.html#K1" title="MCART_6:func.1">[</a><span class="default"><font color="Maroon" title="c12">x1</font>,<font color="Maroon" title="c13">x2</font>,<font color="Maroon" title="c14">x3</font>,<font color="Maroon" title="c15">x4</font>,<font color="Maroon" title="c16">x5</font>,<font color="Maroon" title="c17">x6</font>,<font color="Maroon" title="c18">x7</font>,<font color="Maroon" title="c19">x8</font>,<font color="Maroon" title="c20">x9</font></span><a href="mcart_6.html#K1" title="MCART_6:func.1">]</a></span> )
 <b>by </b><i><a class="txt" href="mcart_6.html#E1:41"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<a NAME="E4:41_1_1"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="mcart_5.html#K1" title="MCART_5:func.1">[</a><span class="default"><font color="Maroon" title="c12">x1</font>,<font color="Maroon" title="c13">x2</font>,<font color="Maroon" title="c14">x3</font>,<font color="Maroon" title="c15">x4</font>,<font color="Maroon" title="c16">x5</font>,<font color="Maroon" title="c17">x6</font>,<font color="Maroon" title="c18">x7</font>,<font color="Maroon" title="c19">x8</font></span><a href="mcart_5.html#K1" title="MCART_5:func.1">]</a></span>,<font color="Maroon" title="c20">x9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="mcart_6.html#E3:41_1_1"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<a NAME="E5:41_1_1"/>
( <span class="p1"><a href="mcart_5.html#K1" title="MCART_5:func.1">[</a><span class="default"><font color="Maroon" title="c12">x1</font>,<font color="Maroon" title="c13">x2</font>,<font color="Maroon" title="c14">x3</font>,<font color="Maroon" title="c15">x4</font>,<font color="Maroon" title="c16">x5</font>,<font color="Maroon" title="c17">x6</font>,<font color="Maroon" title="c18">x7</font>,<font color="Maroon" title="c19">x8</font></span><a href="mcart_5.html#K1" title="MCART_5:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_5.html#K2" title="MCART_5:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">X3</font>,<font color="Maroon" title="c5">X4</font>,<font color="Maroon" title="c6">X5</font>,<font color="Maroon" title="c7">X6</font>,<font color="Maroon" title="c8">X7</font>,<font color="Maroon" title="c9">X8</font></span><a href="mcart_5.html#K2" title="MCART_5:func.2">:]</a></span> &amp; <font color="Maroon" title="c20">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X9</font> )
 
<b>by </b><i><a class="txt" href="mcart_6.html#E3:41_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="mcart_5.html#T40" title="MCART_5:th.40">MCART_5:40</a></i>;<br/>
<b>hence </b><a NAME="E6:41_1_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="mcart_5.html#K2" title="MCART_5:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">X3</font>,<font color="Maroon" title="c5">X4</font>,<font color="Maroon" title="c6">X5</font>,<font color="Maroon" title="c7">X6</font>,<font color="Maroon" title="c8">X7</font>,<font color="Maroon" title="c9">X8</font></span><a href="mcart_5.html#K2" title="MCART_5:func.2">:]</a></span>,<font color="Maroon" title="c10">X9</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="mcart_6.html#E4:41_1_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:41_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="mcart_5.html#K2" title="MCART_5:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">X3</font>,<font color="Maroon" title="c5">X4</font>,<font color="Maroon" title="c6">X5</font>,<font color="Maroon" title="c7">X6</font>,<font color="Maroon" title="c8">X7</font>,<font color="Maroon" title="c9">X8</font></span><a href="mcart_5.html#K2" title="MCART_5:func.2">:]</a></span>,<font color="Maroon" title="c10">X9</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c12">x12345678</font>, <font color="Maroon" title="c13">x9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:41_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c12">x12345678</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_5.html#K2" title="MCART_5:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">X3</font>,<font color="Maroon" title="c5">X4</font>,<font color="Maroon" title="c6">X5</font>,<font color="Maroon" title="c7">X6</font>,<font color="Maroon" title="c8">X7</font>,<font color="Maroon" title="c9">X8</font></span><a href="mcart_5.html#K2" title="MCART_5:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E5:41_1"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c13">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X9</font>
 <b>and </b><br/><a NAME="E6:41_1"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x12345678</font>,<font color="Maroon" title="c13">x9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">x1</font>, <font color="Maroon" title="c15">x2</font>, <font color="Maroon" title="c16">x3</font>, <font color="Maroon" title="c17">x4</font>, <font color="Maroon" title="c18">x5</font>, <font color="Maroon" title="c19">x6</font>, <font color="Maroon" title="c20">x7</font>, <font color="Maroon" title="c21">x8</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:41_1"/><i><font color="Green" title="E24">A7</font></i>: 
( <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X1</font> &amp; <font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font> &amp; <font color="Maroon" title="c16">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X3</font> &amp; <font color="Maroon" title="c17">x4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X4</font> &amp; <font color="Maroon" title="c18">x5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X5</font> &amp; <font color="Maroon" title="c19">x6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X6</font> &amp; <font color="Maroon" title="c20">x7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X7</font> &amp; <font color="Maroon" title="c21">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X8</font> &amp; <font color="Maroon" title="c12">x12345678</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_5.html#K1" title="MCART_5:func.1">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font>,<font color="Maroon" title="c16">x3</font>,<font color="Maroon" title="c17">x4</font>,<font color="Maroon" title="c18">x5</font>,<font color="Maroon" title="c19">x6</font>,<font color="Maroon" title="c20">x7</font>,<font color="Maroon" title="c21">x8</font></span><a href="mcart_5.html#K1" title="MCART_5:func.1">]</a></span> )
 <b>by </b><i><a class="txt" href="mcart_6.html#E4:41_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="mcart_5.html#T39" title="MCART_5:th.39">MCART_5:39</a></i>;<br/><a NAME="E9:41_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_6.html#K1" title="MCART_6:func.1">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font>,<font color="Maroon" title="c16">x3</font>,<font color="Maroon" title="c17">x4</font>,<font color="Maroon" title="c18">x5</font>,<font color="Maroon" title="c19">x6</font>,<font color="Maroon" title="c20">x7</font>,<font color="Maroon" title="c21">x8</font>,<font color="Maroon" title="c13">x9</font></span><a href="mcart_6.html#K1" title="MCART_6:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="mcart_6.html#E6:41_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="mcart_6.html#E8:41_1"><i><font color="Green" title="E24">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E10:41_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 <b>by </b><i><a class="txt" href="mcart_6.html#E1:41"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="mcart_6.html#E5:41_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="mcart_6.html#E8:41_1"><i><font color="Green" title="E24">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:41"/><b>then </b>
<font color="Maroon" title="c1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="mcart_5.html#K2" title="MCART_5:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">X3</font>,<font color="Maroon" title="c5">X4</font>,<font color="Maroon" title="c6">X5</font>,<font color="Maroon" title="c7">X6</font>,<font color="Maroon" title="c8">X7</font>,<font color="Maroon" title="c9">X8</font></span><a href="mcart_5.html#K2" title="MCART_5:func.2">:]</a></span>,<font color="Maroon" title="c10">X9</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E4:41"/>
<font color="Maroon" title="c1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_6.html#K2" title="MCART_6:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c4">X3</font>,<font color="Maroon" title="c5">X4</font>,<font color="Maroon" title="c6">X5</font>,<font color="Maroon" title="c7">X6</font>,<font color="Maroon" title="c8">X7</font>,<font color="Maroon" title="c9">X8</font>,<font color="Maroon" title="c10">X9</font></span><a href="mcart_6.html#K2" title="MCART_6:func.2">:]</a></span>
 ;<br/>


</div>
