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<span class="kw">let </span><font color="Maroon" title="c1">E</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">d</font>, <font color="Olive" title="b3">d9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  st <font color="Olive" title="b3">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">d</font> &amp; <font color="Olive" title="b2">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Olive" title="b1">A</font>) holds <br/>( <font color="Olive" title="b3">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Olive" title="b1">A</font>) &amp;  ex <font color="Olive" title="b4">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b4">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b3">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Olive" title="b4">B</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">d</font>, <font color="Olive" title="b2">d9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  st <font color="Olive" title="b2">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>) holds <br/>( <font color="Olive" title="b2">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>) &amp;  ex <font color="Olive" title="b3">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b3">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Olive" title="b3">B</font>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">d</font>, <font color="Maroon" title="c4">d9</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</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">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c3">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>) implies ( <font color="Maroon" title="c4">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>) &amp;  ex <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Olive" title="b1">B</font>) ) ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c4">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">d</font>
 <span class="kw">and </span><br/><a NAME="E2:7"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</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">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>) &amp;  ex <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Olive" title="b1">B</font>) ) )</span><br/>

<a NAME="E3:7"/>
<font color="Maroon" title="c3">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">d1</font> where <font color="Olive" title="b1">d1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> :  for <font color="Olive" title="b2">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  st <font color="Olive" title="b2">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">d1</font> holds <br/> ex <font color="Olive" title="b3">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b3">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Olive" title="b3">B</font>) ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="zf_colla.html#T1" target="_self" onmouseover="rs('zf_colla/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E4:7"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d1</font> &amp; (  for <font color="Olive" title="b2">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>  st <font color="Olive" title="b2">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">d1</font> holds <br/> ex <font color="Olive" title="b3">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b3">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Olive" title="b3">B</font>) ) ) )
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E6:7"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E7:7"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c4">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c5">B</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/>
<a NAME="E8:7"/>
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c5">B</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_colla.html#T4" target="_self" onmouseover="rs('zf_colla/T4')" onmouseout="rh()">Th4</a>, <a class="txt" href="#E6:7"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:7"/>
<font color="Maroon" title="c4">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">A</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7"><span class="lab"><font color="Green" title="E9">A4</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">  ex <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Olive" title="b1">B</font>) )</span><br/>

<span class="kw">thus </span><a NAME="E10:7"/>
 ex <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">d9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Olive" title="b1">B</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E7:7"><span class="lab"><font color="Green" title="E9">A4</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/>


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