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<span class="kw">let </span><font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><br/>  for <font color="Olive" title="b2">U</font> being   <a href="classes3.html#NM1" title="CLASSES3:NM.1">Grothendieck</a>  st <font color="Maroon" title="c1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">U</font> &amp; <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">A</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  holds <br/><font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">U</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</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" title="b1">U</font> being   <a href="classes3.html#NM1" title="CLASSES3:NM.1">Grothendieck</a>  st <font color="Maroon" title="c1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  holds <br/><font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">U</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  implies <font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">U</font> )</span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>] <span class="kw">means </span> 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>,$1 <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  &amp; <font color="Olive" title="b1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">U</font> holds <br/>$1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">U</font>;<br/>
<a NAME="E1:24"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st (  for <font color="Olive" title="b2">C</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b2">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">C</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</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" title="b1">C</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b1">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">A</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
 for <font color="Olive" title="b1">C</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b1">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</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">S<sub>1</sub></font>[<font color="Maroon" title="c4">A</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">S</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"> ( <font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c4">A</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  &amp; <font color="Maroon" title="c5">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">U</font> implies <font color="Maroon" title="c4">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">U</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:24_1"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
( <font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c4">A</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  &amp; <font color="Maroon" title="c5">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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="c4">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">U</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">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="E4:24_1"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
( <font color="Maroon" title="c6">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="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="wellord2.html#D4" onmouseover="rs('wellord2/D4')" onmouseout="rh()">WELLORD2:def 4</a></span>;<br/>
<a NAME="E5:24_1"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="24_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</font> or <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1_1"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">f</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="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">B</font> = <font color="Maroon" title="c7">y</font> as   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a></span>;<br/>
<a NAME="E3:24_1_1"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="relat_1.html#K6" title="RELAT_1:func.6">|`</a> <font color="Maroon" title="c6">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="funct_1.html#T58" onmouseover="rs('funct_1/T58')" onmouseout="rh()">FUNCT_1:58</a></span>;<br/>
<a NAME="E4:24_1_1"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="relat_1.html#K6" title="RELAT_1:func.6">|`</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="relat_1.html#T89" onmouseover="rs('relat_1/T89')" onmouseout="rh()">RELAT_1:89</a>, <a class="txt" href="#E1:24_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E5:24_1_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="relat_1.html#K6" title="RELAT_1:func.6">|`</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="classes1.html#D1" onmouseover="rs('classes1/D1')" onmouseout="rh()">CLASSES1:def 1</a>, <a class="ref" href="relat_1.html#T186" onmouseover="rs('relat_1/T186')" onmouseout="rh()">RELAT_1:186</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:24_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1_1"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E3:24_1_1"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="wellord2.html#D4" onmouseover="rs('wellord2/D4')" onmouseout="rh()">WELLORD2:def 4</a>, <a class="txt" href="#E1:24_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E11">A2</font></span></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="E6:24_1"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E2:24_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="classes3.html#T2" target="_self" onmouseover="rs('classes3/T2')" onmouseout="rh()">Th2</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>
<a NAME="E2:24"/>
 for <font color="Olive" title="b1">O</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal1.html#S2" onmouseover="rs('ordinal1/S2')" onmouseout="rh()">ORDINAL1:sch 2</a>(<a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E10">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E3:24"/>
( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> <a href="tarski.html#R3" title="TARSKI:pred.3">are_equipotent</a>  implies <font color="Maroon" title="c2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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/>


</div>
