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<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> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</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></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> implies  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">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="b2">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="b3">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="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> ) )</span><br/>







<b>assume </b><a NAME="E1:69"/>
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</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></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">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="b2">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="b3">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="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> )</span><br/>

<a NAME="E2:69"/><b>then </b>
<font color="Maroon" title="c1">z</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="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">X3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D3" title="ZFMISC_1:def.3">ZFMISC_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">x12</font>, <font color="Maroon" title="c6">x3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:69"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c5">x12</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"><font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c3">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E5:69"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c6">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X3</font>
 <b>and </b><br/><a NAME="E6:69"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c1">z</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="c5">x12</font>,<font color="Maroon" title="c6">x3</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="c7">x1</font>, <font color="Maroon" title="c8">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:69"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X1</font>
 <b>and </b><br/><a NAME="E9:69"/><i><font color="Green" title="E47">A5</font></i>: 
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X2</font>
 <b>and </b><br/><a NAME="E10:69"/><i><font color="Green" title="E48">A6</font></i>: 
<font color="Maroon" title="c5">x12</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="c7">x1</font>,<font color="Maroon" title="c8">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="mcart_1.html#E4:69"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E11:69"/>
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c6">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<b>by </b><i><a class="txt" href="mcart_1.html#E6:69"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="mcart_1.html#E10:69"><i><font color="Green" title="E48">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:69"/>
 ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">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="b2">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="b3">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="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> )
 <b>by </b><i><a class="txt" href="mcart_1.html#E5:69"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="mcart_1.html#E8:69"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="mcart_1.html#E9:69"><i><font color="Green" title="E47">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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