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

<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">  ex <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</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> holds  <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Olive" title="b2">d</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>,  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>,  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> ] <span class="kw">means </span>(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> ($3,$1) &amp; (  for <font color="Olive" title="b1">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="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> ($3,$1) holds <br/>$2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Olive" title="b1">d</font> ) );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>] <span class="kw">means </span> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon">S<sub>1</sub></font>[$1,<font color="Olive" title="b1">f1</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[$1,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">E</font>] holds <br/><font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <font color="Maroon">S<sub>1</sub></font>[$1,<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">E</font>];<br/>
<a NAME="E1:10"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><br/>  for <font color="Olive" title="b3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">f</font>,<font color="Maroon" title="c1">E</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><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>)</span>)</span>,<font color="Maroon" title="c1">E</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">E</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><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>)</span>,<font color="Maroon" title="c1">E</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">E</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><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>)</span>,<font color="Maroon" title="c1">E</font>] )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E2:10_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">B</font>)
 <span class="kw">and </span><br/><a NAME="E3:10_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
 for <font color="Olive" title="b1">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="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="c3">B</font>) holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Olive" title="b1">d</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="c2">A</font>,<font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><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>)</span>,<font color="Maroon" title="c1">E</font>]</span><br/>

<a NAME="E4:10_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
 <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 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="c3">B</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="zf_colla.html#T4" target="_self" onmouseover="rs('zf_colla/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:10_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E1:10_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="zf_colla.html#T4" target="_self" onmouseover="rs('zf_colla/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</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" title="b1">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="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/><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Olive" title="b1">d</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">d</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="c5">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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E6:10_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c5">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font></span><br/>

<a NAME="E7:10_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font> )</span><br/>

<a NAME="E1:10_1_1"/><span class="lab"><font color="Green" title="E15">A7</font></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">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E1:10_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="zf_colla.html#T4" target="_self" onmouseover="rs('zf_colla/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:10_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:10_1_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E5:10_1_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">and </span><br/><a NAME="E6:10_1_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E7:10_1_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="zf_colla.html#T7" target="_self" onmouseover="rs('zf_colla/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">d1</font> = <font color="Maroon" title="c7">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a></span>;<br/>
<a NAME="E9:10_1_1"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c8">d1</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="#E6:10_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E5:10_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="zf_colla.html#T6" target="_self" onmouseover="rs('zf_colla/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E10:10_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:10_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E6:10_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E9:10_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E1:10_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</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="E8:10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A12</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font>
 
;<br/>
<a NAME="E9:10_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T128" onmouseover="rs('relat_1/T128')" onmouseout="rh()">RELAT_1:128</a></span>;<br/>
<a NAME="E10:10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A13</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:10_1"><span class="lab"><font color="Green" title="E15">A12</font></span></a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> = 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><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>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E4:10_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:10_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E10:10_1"><span class="lab"><font color="Green" title="E16">A13</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>
<div><span class="lab"><font color="Green" title="E10">A14</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_2"><span class="kw">now </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" title="b1">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st (  for <font color="Olive" title="b2">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b2">B</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>2</sub></font>[<font color="Olive" title="b2">B</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">A</font>]</span></a><div class="add"><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">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <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> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>] ) implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">A</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:10_2"/><span class="lab"><font color="Green" title="E11">A15</font></span>: 
 for <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <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> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</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>2</sub></font>[<font color="Maroon" title="c2">A</font>]</span><br/><span class="kw">thus </span><a NAME="E2:10_2"/>
<font color="Maroon">S<sub>2</sub></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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">E</font>] implies <font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10_2_1"/><span class="lab"><font color="Green" title="E12">A16</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c1">E</font>]
 <span class="kw">and </span><br/><a NAME="E2:10_2_1"/><span class="lab"><font color="Green" title="E13">A17</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">f2</font>,<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="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_2_1_1"><span class="kw">now </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" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN: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="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">x</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:10_2_1_1"/><span class="lab"><font color="Green" title="E14">A18</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span><br/><a NAME="E2:10_2_1_1"/>
 <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 href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_colla.html#T7" target="_self" onmouseover="rs('zf_colla/T7')" onmouseout="rh()">Th7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">d</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_1_1"><span class="lab"><font color="Green" title="E14">A18</font></span></a></span>;<br/><a NAME="E4:10_2_1_1"/><span class="lab"><font color="Green" title="E15">A19</font></span>: 
<font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:10_2_1_1_1"/>
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</font> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</font>
  <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="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_2_1_1_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> ; <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">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</font> implies <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_2_1_1_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">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</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="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:10_2_1_1_1_1"/><span class="lab"><font color="Green" title="E16">A20</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f1</font>
 <span class="kw">and </span><br/><a NAME="E4:10_2_1_1_1_1"/><span class="lab"><font color="Green" title="E17">A21</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">d</font>
 <span class="kw">and </span><br/><a NAME="E5:10_2_1_1_1_1"/><span class="lab"><font color="Green" title="E18">A22</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E6:10_2_1_1_1_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_1"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="ref" href="zf_colla.html#T7" target="_self" onmouseover="rs('zf_colla/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">d1</font> = <font color="Maroon" title="c8">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_2_1_1_1_1"><span class="lab"><font color="Green" title="E16">A20</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E9:10_2_1_1_1_1"/><span class="lab"><font color="Green" title="E19">A23</font></span>: 
<font color="Maroon" title="c10">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="E10:10_2_1_1_1_1"/><span class="lab"><font color="Green" title="E20">A24</font></span>: 
<font color="Maroon" title="c9">d1</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="c10">B</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_1_1"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="txt" href="#E4:10_2_1_1_1_1"><span class="lab"><font color="Green" title="E17">A21</font></span></a>, <a class="ref" href="zf_colla.html#T6" target="_self" onmouseover="rs('zf_colla/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E11:10_2_1_1_1_1"/>
<font color="Maroon" title="c10">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:10_2_1_1_1_1"><span class="lab"><font color="Green" title="E19">A23</font></span></a>, <a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E12:10_2_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">B</font>,<font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font>)</span>)</span>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">B</font>,<font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font>)</span>)</span>,<font color="Maroon" title="c1">E</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E1:10_2_1"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="txt" href="#E2:10_2_1"><span class="lab"><font color="Green" title="E13">A17</font></span></a></span>;<br/>
<a NAME="E13:10_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A25</font></span>: 
<font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2"><span class="lab"><font color="Green" title="E11">A15</font></span></a>, <a class="txt" href="#E9:10_2_1_1_1_1"><span class="lab"><font color="Green" title="E19">A23</font></span></a></span>;<br/>
<font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:10_2_1_1_1_1"><span class="lab"><font color="Green" title="E20">A24</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:10_2_1_1_1_1"><span class="lab"><font color="Green" title="E20">A24</font></span></a>, <a class="txt" href="#E13:10_2_1_1_1_1"><span class="lab"><font color="Green" title="E21">A25</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:10_2_1_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_1"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="txt" href="#E2:10_2_1"><span class="lab"><font color="Green" title="E13">A17</font></span></a>, <a class="txt" href="#E3:10_2_1_1_1_1"><span class="lab"><font color="Green" title="E16">A20</font></span></a>, <a class="txt" href="#E4:10_2_1_1_1_1"><span class="lab"><font color="Green" title="E17">A21</font></span></a>, <a class="txt" href="#E5:10_2_1_1_1_1"><span class="lab"><font color="Green" title="E18">A22</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</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">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> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</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">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:10_2_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</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">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:10_2_1_1_1"/><span class="lab"><font color="Green" title="E16">A26</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f2</font>
 <span class="kw">and </span><br/><a NAME="E5:10_2_1_1_1"/><span class="lab"><font color="Green" title="E17">A27</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">d</font>
 <span class="kw">and </span><br/><a NAME="E6:10_2_1_1_1"/><span class="lab"><font color="Green" title="E18">A28</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E7:10_2_1_1_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_2_1"><span class="lab"><font color="Green" title="E13">A17</font></span></a>, <a class="ref" href="zf_colla.html#T7" target="_self" onmouseover="rs('zf_colla/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">d1</font> = <font color="Maroon" title="c8">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_2_1_1_1"><span class="lab"><font color="Green" title="E16">A26</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E10:10_2_1_1_1"/><span class="lab"><font color="Green" title="E19">A29</font></span>: 
<font color="Maroon" title="c10">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="E11:10_2_1_1_1"/><span class="lab"><font color="Green" title="E20">A30</font></span>: 
<font color="Maroon" title="c9">d1</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="c10">B</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_1_1"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="txt" href="#E5:10_2_1_1_1"><span class="lab"><font color="Green" title="E17">A27</font></span></a>, <a class="ref" href="zf_colla.html#T6" target="_self" onmouseover="rs('zf_colla/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E12:10_2_1_1_1"/>
<font color="Maroon" title="c10">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:10_2_1_1_1"><span class="lab"><font color="Green" title="E19">A29</font></span></a>, <a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E13:10_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">B</font>,<font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font>)</span>)</span>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">B</font>,<font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font>)</span>)</span>,<font color="Maroon" title="c1">E</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E1:10_2_1"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="txt" href="#E2:10_2_1"><span class="lab"><font color="Green" title="E13">A17</font></span></a></span>;<br/>
<a NAME="E14:10_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A31</font></span>: 
<font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2"><span class="lab"><font color="Green" title="E11">A15</font></span></a>, <a class="txt" href="#E10:10_2_1_1_1"><span class="lab"><font color="Green" title="E19">A29</font></span></a></span>;<br/>
<font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_2_1_1_1"><span class="lab"><font color="Green" title="E20">A30</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_2_1_1_1"><span class="lab"><font color="Green" title="E20">A30</font></span></a>, <a class="txt" href="#E14:10_2_1_1_1"><span class="lab"><font color="Green" title="E21">A31</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:10_2_1_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">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_1"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="txt" href="#E2:10_2_1"><span class="lab"><font color="Green" title="E13">A17</font></span></a>, <a class="txt" href="#E4:10_2_1_1_1"><span class="lab"><font color="Green" title="E16">A26</font></span></a>, <a class="txt" href="#E5:10_2_1_1_1"><span class="lab"><font color="Green" title="E17">A27</font></span></a>, <a class="txt" href="#E6:10_2_1_1_1"><span class="lab"><font color="Green" title="E18">A28</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</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="E5:10_2_1_1"/>
<font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_1"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="txt" href="#E1:10_2_1_1"><span class="lab"><font color="Green" title="E14">A18</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:10_2_1_1"/>
<font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_2_1"><span class="lab"><font color="Green" title="E13">A17</font></span></a>, <a class="txt" href="#E1:10_2_1_1"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="txt" href="#E4:10_2_1_1"><span class="lab"><font color="Green" title="E15">A19</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="E4:10_2_1"/>
<font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_1"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="txt" href="#E2:10_2_1"><span class="lab"><font color="Green" title="E13">A17</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</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></div><span class="kw">end;</span></div>
<a NAME="E3:10"/><span class="lab"><font color="Green" title="E11">A32</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">A</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="#E2:10"><span class="lab"><font color="Green" title="E10">A14</font></span></a>);<br/></span>
<a NAME="E4:10"/><span class="lab"><font color="Green" title="E12">A33</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">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <font color="Olive" title="b2">B</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>3</sub></font>[<font color="Olive" title="b2">B</font>] ) holds <br/><font color="Maroon">S<sub>3</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="10_3"><span class="kw">proof </span></a><div class="add">

<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">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <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> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">B</font>] ) implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">A</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_3"/>
 for <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>  st <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> holds <br/> ex <font color="Olive" title="b2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">f</font>,<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">S<sub>3</sub></font>[<font color="Maroon" title="c2">A</font>]</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">d</font>, <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">e</font> iff  ex <font color="Olive" title="b4">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b5">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> ex <font color="Olive" title="b6">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b4">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b5">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="b4">d1</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="b5">B</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b5">B</font>,<font color="Olive" title="b6">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">d1</font> ) ) ) );<br/>
<a NAME="E2:10_3"/><span class="lab"><font color="Green" title="E13">A34</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN: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/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:10_3_1"/><span class="lab"><font color="Green" title="E14">A35</font></span>: 
 <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 href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_colla.html#T7" target="_self" onmouseover="rs('zf_colla/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN: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  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E2:10_3_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN: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">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">d</font> = <font color="Maroon" title="c3">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_3_1"><span class="lab"><font color="Green" title="E14">A35</font></span></a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> ex <font color="Olive" title="b2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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; $1 <a href="hidden.html#R2" title="HIDDEN: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>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">E</font>] &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E4:10_3_1"/><span class="lab"><font color="Green" title="E15">A36</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">S<sub>5</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font>] implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c8">B1</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>, <font color="Maroon" title="c9">f1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><a NAME="E2:10_3_1_1"/>
<font color="Maroon" title="c8">B1</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="E3:10_3_1_1"/><span class="lab"><font color="Green" title="E16">A37</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN: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="c8">B1</font>)
 <span class="kw">and </span><br/><a NAME="E4:10_3_1_1"/><span class="lab"><font color="Green" title="E17">A38</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">B1</font>,<font color="Maroon" title="c9">f1</font>,<font color="Maroon" title="c1">E</font>]
 <span class="kw">and </span><br/><a NAME="E5:10_3_1_1"/><span class="lab"><font color="Green" title="E18">A39</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</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"> (  not <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font>] or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c10">B2</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>, <font color="Maroon" title="c11">f2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><a NAME="E7:10_3_1_1"/>
<font color="Maroon" title="c10">B2</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="E8:10_3_1_1"/><span class="lab"><font color="Green" title="E19">A40</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN: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="c10">B2</font>)
 <span class="kw">and </span><br/><a NAME="E9:10_3_1_1"/><span class="lab"><font color="Green" title="E20">A41</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">B2</font>,<font color="Maroon" title="c11">f2</font>,<font color="Maroon" title="c1">E</font>]
 <span class="kw">and </span><br/><a NAME="E10:10_3_1_1"/><span class="lab"><font color="Green" title="E21">A42</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</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="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font></span><br/>

<div><span class="lab"><font color="Green" title="E22">A43</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_1_1_1"><span class="kw">now </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"> ( <font color="Maroon" title="c10">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">B1</font> implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:10_3_1_1_1"/>
<font color="Maroon" title="c10">B2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">B1</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="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font></span><br/><a NAME="E2:10_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">B2</font>,<font color="Maroon" title="c9">f1</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B2</font>)</span>)</span>,<font color="Maroon" title="c1">E</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E4:10_3_1_1"><span class="lab"><font color="Green" title="E17">A38</font></span></a></span>;<br/><a NAME="E3:10_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">f1</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c10">B2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A32</font></span></a>, <a class="txt" href="#E9:10_3_1_1"><span class="lab"><font color="Green" title="E20">A41</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:10_3_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_3_1_1"><span class="lab"><font color="Green" title="E18">A39</font></span></a>, <a class="txt" href="#E8:10_3_1_1"><span class="lab"><font color="Green" title="E19">A40</font></span></a>, <a class="txt" href="#E10:10_3_1_1"><span class="lab"><font color="Green" title="E21">A42</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</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>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_1_1_2"><span class="kw">now </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"> ( <font color="Maroon" title="c8">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">B2</font> implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:10_3_1_1_2"/>
<font color="Maroon" title="c8">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c10">B2</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="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font></span><br/><a NAME="E2:10_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">B1</font>,<font color="Maroon" title="c11">f2</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c8">B1</font>)</span>)</span>,<font color="Maroon" title="c1">E</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E9:10_3_1_1"><span class="lab"><font color="Green" title="E20">A41</font></span></a></span>;<br/><a NAME="E3:10_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">f2</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c8">B1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A32</font></span></a>, <a class="txt" href="#E4:10_3_1_1"><span class="lab"><font color="Green" title="E17">A38</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:10_3_1_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_3_1_1"><span class="lab"><font color="Green" title="E16">A37</font></span></a>, <a class="txt" href="#E5:10_3_1_1"><span class="lab"><font color="Green" title="E18">A39</font></span></a>, <a class="txt" href="#E10:10_3_1_1"><span class="lab"><font color="Green" title="E21">A42</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</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="E13:10_3_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_3_1_1"><span class="lab"><font color="Green" title="E22">A43</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">consider </span><font color="Maroon" title="c5">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E6:10_3_1"/><span class="lab"><font color="Green" title="E16">A44</font></span>: 
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c5">X</font> iff  ex <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="tarski.html#S1" onmouseover="rs('tarski/S1')" onmouseout="rh()">TARSKI:sch 1</a>(<a class="txt" href="#E4:10_3_1"><span class="lab"><font color="Green" title="E15">A36</font></span></a>);<br/></span>
<span class="kw">take </span><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">X</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>4</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">y</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c4">d</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">  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">e</font> iff  ex <font color="Olive" title="b3">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b4">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> ex <font color="Olive" title="b5">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b3">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Olive" title="b4">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="b3">d1</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>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b4">B</font>,<font color="Olive" title="b5">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">d1</font> ) ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">X</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">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> iff  ex <font color="Olive" title="b2">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b3">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> ex <font color="Olive" title="b4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b2">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <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">d1</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>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">B</font>,<font color="Olive" title="b4">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">d1</font> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:10_3_1"/>
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</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="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> iff  ex <font color="Olive" title="b2">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b3">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> ex <font color="Olive" title="b4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b2">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <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">d1</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>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">B</font>,<font color="Olive" title="b4">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">d1</font> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E8:10_3_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">X</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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> iff  ex <font color="Olive" title="b2">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b3">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> ex <font color="Olive" title="b4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b2">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <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">d1</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>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">B</font>,<font color="Olive" title="b4">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">d1</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> iff  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> ex <font color="Olive" title="b2">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> ex <font color="Olive" title="b3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Olive" title="b2">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="b1">d1</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="b2">B</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d1</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E9:10_3_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> implies  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> ex <font color="Olive" title="b2">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> ex <font color="Olive" title="b3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Olive" title="b2">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="b1">d1</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="b2">B</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d1</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"> (  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> ex <font color="Olive" title="b2">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> ex <font color="Olive" title="b3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Olive" title="b2">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="b1">d1</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="b2">B</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d1</font> ) implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:10_3_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</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">  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> ex <font color="Olive" title="b2">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> ex <font color="Olive" title="b3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Olive" title="b2">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="b1">d1</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="b2">B</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d1</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:10_3_1_2"/><span class="lab"><font color="Green" title="E17">A45</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">and </span><br/><a NAME="E4:10_3_1_2"/><span class="lab"><font color="Green" title="E18">A46</font></span>: 
 ex <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> ex <font color="Olive" title="b2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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="c8">y</font> <a href="hidden.html#R2" title="HIDDEN: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>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10_3_1"><span class="lab"><font color="Green" title="E16">A44</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>, <font color="Maroon" title="c10">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="E6:10_3_1_2"/><span class="lab"><font color="Green" title="E19">A47</font></span>: 
<font color="Maroon" title="c9">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="E7:10_3_1_2"/><span class="lab"><font color="Green" title="E20">A48</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN: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="c9">B</font>)
 <span class="kw">and </span><br/><a NAME="E8:10_3_1_2"/><span class="lab"><font color="Green" title="E21">A49</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_3_1_2"><span class="lab"><font color="Green" title="E18">A46</font></span></a></span>;<br/>
<a NAME="E9:10_3_1_2"/>
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c9">B</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_colla.html#T7" target="_self" onmouseover="rs('zf_colla/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">d1</font> = <font color="Maroon" title="c8">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10_3_1_2"><span class="lab"><font color="Green" title="E20">A48</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">d1</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">  ex <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> ex <font color="Olive" title="b2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c11">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <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="c11">d1</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>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">d1</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c9">B</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">  ex <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c11">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c9">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="c11">d1</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="c9">B</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B</font>,<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">d1</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c10">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="c11">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c9">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="c11">d1</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="c9">B</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">d1</font> )</span><br/>

<span class="kw">thus </span><a NAME="E11:10_3_1_2"/>
( <font color="Maroon" title="c11">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c9">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="c11">d1</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="c9">B</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">d1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_3_1_2"><span class="lab"><font color="Green" title="E17">A45</font></span></a>, <a class="txt" href="#E6:10_3_1_2"><span class="lab"><font color="Green" title="E19">A47</font></span></a>, <a class="txt" href="#E7:10_3_1_2"><span class="lab"><font color="Green" title="E20">A48</font></span></a>, <a class="txt" href="#E8:10_3_1_2"><span class="lab"><font color="Green" title="E21">A49</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">given </span><font color="Maroon" title="c8">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c9">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>, <font color="Maroon" title="c10">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><a NAME="E11:10_3_1"/><span class="lab"><font color="Green" title="E17">A50</font></span>: 
( <font color="Maroon" title="c8">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c9">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="c8">d1</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="c9">B</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">d1</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font></span><br/>

<span class="kw">thus </span><a NAME="E12:10_3_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10_3_1"><span class="lab"><font color="Green" title="E16">A44</font></span></a>, <a class="txt" href="#E11:10_3_1"><span class="lab"><font color="Green" title="E17">A50</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">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="E4:10_3"/><span class="lab"><font color="Green" title="E14">A51</font></span>: 
(  <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="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; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN: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="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" onmouseover="rs('classes1/S1')" onmouseout="rh()">CLASSES1:sch 1</a>(<a class="txt" href="#E2:10_3"><span class="lab"><font color="Green" title="E13">A34</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>] <span class="kw">means </span>( $1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font> implies <font color="Maroon">S<sub>1</sub></font>[$1,<font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,$1)</span>)</span>,<font color="Maroon" title="c1">E</font>] );<br/>
<a NAME="E5:10_3"/><span class="lab"><font color="Green" title="E15">A52</font></span>: 
 for <font color="Olive" title="b1">B</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">B</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b2">C</font>] ) holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">B</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">B</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">B</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">C</font>] ) implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">B</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_3_2"/><span class="lab"><font color="Green" title="E16">A53</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">B</font> holds <br/><font color="Maroon">S<sub>5</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>5</sub></font>[<font color="Maroon" title="c4">B</font>]</span><br/>

<span class="kw">assume </span><a NAME="E2:10_3_2"/><span class="lab"><font color="Green" title="E17">A54</font></span>: 
<font color="Maroon" title="c4">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <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">S<sub>1</sub></font>[<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span>,<font color="Maroon" title="c1">E</font>]</span><br/>

<a NAME="E3:10_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A55</font></span>: 
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">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></span>;<br/>
<span class="kw">thus </span><a NAME="E4:10_3_2"/><span class="lab"><font color="Green" title="E19">A56</font></span>: 
 <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="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">B</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_3"><span class="lab"><font color="Green" title="E14">A51</font></span></a>, <a class="txt" href="#E2:10_3_2"><span class="lab"><font color="Green" title="E17">A54</font></span></a>, <a class="ref" href="zf_colla.html#T4" target="_self" onmouseover="rs('zf_colla/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</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" title="b1">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="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="c4">B</font>) holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Olive" title="b1">d</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">d</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="c5">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="c4">B</font>) implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:10_3_2"/><span class="lab"><font color="Green" title="E20">A57</font></span>: 
<font color="Maroon" title="c5">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="c4">B</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font></span><br/>

<a NAME="E6:10_3_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">d9</font>, <font color="Maroon" title="c7">e</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E8:10_3_2"/><span class="lab"><font color="Green" title="E21">A58</font></span>: 
<font color="Maroon" title="c6">d9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">and </span><br/><a NAME="E9:10_3_2"/><span class="lab"><font color="Green" title="E22">A59</font></span>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">and </span><br/><a NAME="E10:10_3_2"/><span class="lab"><font color="Green" title="E23">A60</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">e</font> iff  ex <font color="Olive" title="b2">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b3">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> ex <font color="Olive" title="b4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b2">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">d9</font> &amp; <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">d1</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>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">B</font>,<font color="Olive" title="b4">f</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">d1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_3"><span class="lab"><font color="Green" title="E14">A51</font></span></a>, <a class="txt" href="#E3:10_3_2"><span class="lab"><font color="Green" title="E18">A55</font></span></a>, <a class="txt" href="#E5:10_3_2"><span class="lab"><font color="Green" title="E20">A57</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">X</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font>;<br/>
<a NAME="E11:10_3_2"/><span class="lab"><font color="Green" title="E24">A61</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_3_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">d1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c11">C</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a>, <font color="Maroon" title="c12">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:10_3_2_1"/><span class="lab"><font color="Green" title="E25">A62</font></span>: 
<font color="Maroon" title="c10">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">d9</font>
 <span class="kw">and </span><br/><a NAME="E4:10_3_2_1"/>
<font color="Maroon" title="c11">C</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="E5:10_3_2_1"/><span class="lab"><font color="Green" title="E26">A63</font></span>: 
<font color="Maroon" title="c10">d1</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="c11">C</font>)
 <span class="kw">and </span><br/><a NAME="E6:10_3_2_1"/><span class="lab"><font color="Green" title="E27">A64</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">C</font>,<font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c1">E</font>]
 <span class="kw">and </span><br/><a NAME="E7:10_3_2_1"/><span class="lab"><font color="Green" title="E28">A65</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:10_3_2"><span class="lab"><font color="Green" title="E23">A60</font></span></a>, <a class="txt" href="#E9:10_3_2"><span class="lab"><font color="Green" title="E22">A59</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">C9</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E9:10_3_2_1"/><span class="lab"><font color="Green" title="E29">A66</font></span>: 
<font color="Maroon" title="c13">C9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E10:10_3_2_1"/><span class="lab"><font color="Green" title="E30">A67</font></span>: 
<font color="Maroon" title="c10">d1</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="c13">C9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_3_2"><span class="lab"><font color="Green" title="E20">A57</font></span></a>, <a class="txt" href="#E8:10_3_2"><span class="lab"><font color="Green" title="E21">A58</font></span></a>, <a class="txt" href="#E3:10_3_2_1"><span class="lab"><font color="Green" title="E25">A62</font></span></a>, <a class="ref" href="zf_colla.html#T6" target="_self" onmouseover="rs('zf_colla/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E11:10_3_2_1"/><span class="lab"><font color="Green" title="E31">A68</font></span>: 
 for <font color="Olive" title="b1">C</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><br/>  for <font color="Olive" title="b2">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c13">C9</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">h</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c10">d1</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">C</font>) &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d1</font> holds <br/><font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">C</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">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon" title="c14">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c13">C9</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">C</font>,<font color="Olive" title="b1">h</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c10">d1</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="c14">C</font>) &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d1</font> holds <br/><font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font></span><br/><span class="kw">let </span><font color="Maroon" title="c15">h</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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="c14">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c13">C9</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">C</font>,<font color="Maroon" title="c15">h</font>,<font color="Maroon" title="c1">E</font>] &amp; <font color="Maroon" title="c10">d1</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="c14">C</font>) &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d1</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10_3_2_1_1"/><span class="lab"><font color="Green" title="E32">A69</font></span>: 
<font color="Maroon" title="c14">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c13">C9</font>
 <span class="kw">and </span><br/><a NAME="E2:10_3_2_1_1"/><span class="lab"><font color="Green" title="E33">A70</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">C</font>,<font color="Maroon" title="c15">h</font>,<font color="Maroon" title="c1">E</font>]
 <span class="kw">and </span><br/><a NAME="E3:10_3_2_1_1"/><span class="lab"><font color="Green" title="E34">A71</font></span>: 
<font color="Maroon" title="c10">d1</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="c14">C</font>)
 <span class="kw">and </span><br/><a NAME="E4:10_3_2_1_1"/><span class="lab"><font color="Green" title="E35">A72</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d1</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font></span><br/>

<a NAME="E5:10_3_2_1_1"/><span class="lab"><font color="Green" title="E36">A73</font></span>: 
<font color="Maroon" title="c14">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:10_3_2_1"><span class="lab"><font color="Green" title="E29">A66</font></span></a>, <a class="txt" href="#E1:10_3_2_1_1"><span class="lab"><font color="Green" title="E32">A69</font></span></a>, <a class="ref" href="ordinal1.html#T12" onmouseover="rs('ordinal1/T12')" onmouseout="rh()">ORDINAL1:12</a></span>;<br/>
<a NAME="E6:10_3_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_3_2"><span class="lab"><font color="Green" title="E17">A54</font></span></a>, <a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E7:10_3_2_1_1"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">C</font>,<font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c14">C</font>)</span>)</span>,<font color="Maroon" title="c1">E</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_3_2"><span class="lab"><font color="Green" title="E16">A53</font></span></a>, <a class="txt" href="#E5:10_3_2_1_1"><span class="lab"><font color="Green" title="E36">A73</font></span></a></span>;<br/>
<a NAME="E8:10_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A74</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c14">C</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A32</font></span></a>, <a class="txt" href="#E2:10_3_2_1_1"><span class="lab"><font color="Green" title="E33">A70</font></span></a></span>;<br/>
<a NAME="E9:10_3_2_1_1"/><span class="lab"><font color="Green" title="E38">A75</font></span>: 
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c14">C</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="c4">B</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="#E5:10_3_2_1_1"><span class="lab"><font color="Green" title="E36">A73</font></span></a>, <a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E10:10_3_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c14">C</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c14">C</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T51" onmouseover="rs('funct_1/T51')" onmouseout="rh()">FUNCT_1:51</a></span>;<br/>
<a NAME="E11:10_3_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_3_2_1_1"><span class="lab"><font color="Green" title="E34">A71</font></span></a>, <a class="txt" href="#E8:10_3_2_1_1"><span class="lab"><font color="Green" title="E37">A74</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:10_3_2_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_3_2"><span class="lab"><font color="Green" title="E19">A56</font></span></a>, <a class="txt" href="#E8:10_3_2"><span class="lab"><font color="Green" title="E21">A58</font></span></a>, <a class="txt" href="#E3:10_3_2_1"><span class="lab"><font color="Green" title="E25">A62</font></span></a>, <a class="txt" href="#E3:10_3_2_1_1"><span class="lab"><font color="Green" title="E34">A71</font></span></a>, <a class="txt" href="#E4:10_3_2_1_1"><span class="lab"><font color="Green" title="E35">A72</font></span></a>, <a class="txt" href="#E9:10_3_2_1_1"><span class="lab"><font color="Green" title="E38">A75</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</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="E12:10_3_2_1"/>
( <font color="Maroon" title="c13">C9</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">C</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:10_3_2_1_2"/>
<font color="Maroon" title="c13">C9</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c11">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" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font></span><br/>

<a NAME="E2:10_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A76</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">C9</font>,<font color="Maroon" title="c12">h</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c13">C9</font>)</span>)</span>,<font color="Maroon" title="c1">E</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E6:10_3_2_1"><span class="lab"><font color="Green" title="E27">A64</font></span></a></span>;<br/>
<a NAME="E3:10_3_2_1_2"/>
<font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c13">C9</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:10_3_2_1"><span class="lab"><font color="Green" title="E30">A67</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:10_3_2_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10_3_2_1"><span class="lab"><font color="Green" title="E28">A65</font></span></a>, <a class="txt" href="#E10:10_3_2_1"><span class="lab"><font color="Green" title="E30">A67</font></span></a>, <a class="txt" href="#E11:10_3_2_1"><span class="lab"><font color="Green" title="E31">A68</font></span></a>, <a class="txt" href="#E2:10_3_2_1_2"><span class="lab"><font color="Green" title="E32">A76</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="E13:10_3_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_3_2_1"><span class="lab"><font color="Green" title="E26">A63</font></span></a>, <a class="txt" href="#E6:10_3_2_1"><span class="lab"><font color="Green" title="E27">A64</font></span></a>, <a class="txt" href="#E7:10_3_2_1"><span class="lab"><font color="Green" title="E28">A65</font></span></a>, <a class="txt" href="#E11:10_3_2_1"><span class="lab"><font color="Green" title="E31">A68</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>
<a NAME="E12:10_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_3_2_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:10_3_2_2"/><span class="lab"><font color="Green" title="E25">A77</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:10_3_2_2"/><span class="lab"><font color="Green" title="E26">A78</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">and </span><br/><a NAME="E5:10_3_2_2"/><span class="lab"><font color="Green" title="E27">A79</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E6:10_3_2_2"/>
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_colla.html#T7" target="_self" onmouseover="rs('zf_colla/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">d1</font> = <font color="Maroon" title="c10">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_3_2"><span class="lab"><font color="Green" title="E19">A56</font></span></a>, <a class="txt" href="#E3:10_3_2_2"><span class="lab"><font color="Green" title="E25">A77</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">C</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E9:10_3_2_2"/><span class="lab"><font color="Green" title="E28">A80</font></span>: 
<font color="Maroon" title="c12">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E10:10_3_2_2"/><span class="lab"><font color="Green" title="E29">A81</font></span>: 
<font color="Maroon" title="c11">d1</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="c12">C</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_3_2"><span class="lab"><font color="Green" title="E20">A57</font></span></a>, <a class="txt" href="#E4:10_3_2_2"><span class="lab"><font color="Green" title="E26">A78</font></span></a>, <a class="ref" href="zf_colla.html#T6" target="_self" onmouseover="rs('zf_colla/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E11:10_3_2_2"/>
 <a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c12">C</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="c4">B</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="#E9:10_3_2_2"><span class="lab"><font color="Green" title="E28">A80</font></span></a>, <a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E12:10_3_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c12">C</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c12">C</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T51" onmouseover="rs('funct_1/T51')" onmouseout="rh()">FUNCT_1:51</a></span>;<br/>
<a NAME="E13:10_3_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A82</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c12">C</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:10_3_2_2"><span class="lab"><font color="Green" title="E29">A81</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E14:10_3_2_2"/>
<font color="Maroon" title="c12">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_3_2"><span class="lab"><font color="Green" title="E17">A54</font></span></a>, <a class="txt" href="#E9:10_3_2_2"><span class="lab"><font color="Green" title="E28">A80</font></span></a>, <a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E15:10_3_2_2"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">C</font>,<font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c12">C</font>)</span>)</span>,<font color="Maroon" title="c1">E</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_3_2"><span class="lab"><font color="Green" title="E16">A53</font></span></a>, <a class="txt" href="#E9:10_3_2_2"><span class="lab"><font color="Green" title="E28">A80</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:10_3_2_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_3_2"><span class="lab"><font color="Green" title="E17">A54</font></span></a>, <a class="txt" href="#E8:10_3_2"><span class="lab"><font color="Green" title="E21">A58</font></span></a>, <a class="txt" href="#E10:10_3_2"><span class="lab"><font color="Green" title="E23">A60</font></span></a>, <a class="txt" href="#E4:10_3_2_2"><span class="lab"><font color="Green" title="E26">A78</font></span></a>, <a class="txt" href="#E5:10_3_2_2"><span class="lab"><font color="Green" title="E27">A79</font></span></a>, <a class="txt" href="#E9:10_3_2_2"><span class="lab"><font color="Green" title="E28">A80</font></span></a>, <a class="txt" href="#E10:10_3_2_2"><span class="lab"><font color="Green" title="E29">A81</font></span></a>, <a class="txt" href="#E13:10_3_2_2"><span class="lab"><font color="Green" title="E30">A82</font></span></a>, <a class="txt" href="#E9:10_3_2"><span class="lab"><font color="Green" title="E22">A59</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="E13:10_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p2">(<span class="default"><a href="zf_colla.html#K1" title="ZF_COLLA:func.1">Collapse</a> (<font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_3_2"><span class="lab"><font color="Green" title="E24">A61</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>
<a NAME="E6:10_3"/><span class="lab"><font color="Green" title="E16">A83</font></span>: 
 for <font color="Olive" title="b1">B</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">B</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="#E5:10_3"><span class="lab"><font color="Green" title="E15">A52</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c3">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">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">E</font>]</span><br/>

<a NAME="E7:10_3"/>
<font color="Maroon" title="c3">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T68" onmouseover="rs('relat_1/T68')" onmouseout="rh()">RELAT_1:68</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:10_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">E</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_3"><span class="lab"><font color="Green" title="E14">A51</font></span></a>, <a class="txt" href="#E6:10_3"><span class="lab"><font color="Green" title="E16">A83</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>
<a NAME="E5:10"/><span class="lab"><font color="Green" title="E13">A84</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">A</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="#E4:10"><span class="lab"><font color="Green" title="E12">A33</font></span></a>);<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c2">A</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E7:10"/><span class="lab"><font color="Green" title="E14">A85</font></span>: 
<font color="Maroon" title="c1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#T8" target="_self" onmouseover="rs('zf_colla/T8')" onmouseout="rh()">Th8</a></span>;<br/>
<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="E9:10"/><span class="lab"><font color="Green" title="E15">A86</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">E</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E13">A84</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">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"> (  <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> <font color="Maroon" title="c1">E</font> &amp; (  for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> holds  <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Olive" title="b1">d</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E10:10"/>
 <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> <font color="Maroon" title="c1">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E14">A85</font></span></a>, <a class="txt" href="#E9:10"><span class="lab"><font color="Green" title="E15">A86</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">  for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> holds  <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Olive" title="b1">d</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">d</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="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c4">d</font></span><br/>

<span class="kw">thus </span><a NAME="E11:10"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E14">A85</font></span></a>, <a class="txt" href="#E9:10"><span class="lab"><font color="Green" title="E15">A86</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>
