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<span class="kw">let </span><font color="Maroon" title="c1">U</font> be   <a href="classes3.html#NM1" title="CLASSES3:NM.1">Grothendieck</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">U</font> is  <a href="classes1.html#V2" title="CLASSES1:attr.2">Tarski</a> </span><br/>

<span class="kw">thus </span><a NAME="E1:29"/>
<font color="Maroon" title="c1">U</font> is  <a href="classes1.html#V1" title="CLASSES1:attr.1">subset-closed</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="classes1.html#D2" onmouseover="rs('classes1/D2')" onmouseout="rh()">CLASSES1:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">U</font> or  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">U</font> ) ) &amp; (  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">U</font> or <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c1">U</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  or <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">U</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:29"/>
 for <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">U</font> holds <br/> <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Olive" title="b1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">U</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="classes3.html#D1" target="_self" onmouseover="rs('classes3/D1')" onmouseout="rh()">Def1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">U</font> or <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c1">U</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  or <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">U</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">U</font> or <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">U</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  or <font color="Maroon" title="c2">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">U</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:29"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">U</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">U</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  or <font color="Maroon" title="c2">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">U</font> )</span><br/>

<a NAME="E4:29"/>
(  not <font color="Maroon" title="c2">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">U</font> implies <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">U</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:29_1"/>
 not <font color="Maroon" title="c2">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">U</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">U</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:29_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
( <font color="Maroon" title="c3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">U</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="classes3.html#T16" target="_self" onmouseover="rs('classes3/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<a NAME="E4:29_1"/>
 not <font color="Maroon" title="c1">U</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">U</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E6:29_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
( <font color="Maroon" title="c4">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">U</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">U</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="classes3.html#T16" target="_self" onmouseover="rs('classes3/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">gf</font> = <font color="Maroon" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>;<br/>
<a NAME="E7:29_1"/>
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">U</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/>
<a NAME="E8:29_1"/><span class="kw">then </span>
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">U</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a>, <a class="ref" href="relat_1.html#T28" onmouseover="rs('relat_1/T28')" onmouseout="rh()">RELAT_1:28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:29_1"/>
<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">U</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E6:29_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="wellord2.html#D4" onmouseover="rs('wellord2/D4')" onmouseout="rh()">WELLORD2:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:29"/>
( <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">U</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  or <font color="Maroon" title="c2">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">U</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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