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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b2">BASSIGN</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span><br/>  for <font color="Olive" title="b3">g</font> being   <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K49" title="MODELC_1:func.49">BASSModel</a> (<font color="Olive" title="b1">R</font>,<font color="Olive" title="b2">BASSIGN</font>)</span>)</span><br/>  for <font color="Olive" title="b4">s0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b4">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b3">g</font> &amp; (  for <font color="Olive" title="b5">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b3">g</font> holds <br/><font color="Olive" title="b5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Olive" title="b3">g</font> ) holds <br/> ex <font color="Olive" title="b5">pai</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Olive" title="b1">R</font> st <br/>( <font color="Olive" title="b5">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s0</font> &amp; (  for <font color="Olive" title="b6">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b5">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b3">g</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">BASSIGN</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span><br/>  for <font color="Olive" title="b2">g</font> being   <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K49" title="MODELC_1:func.49">BASSModel</a> (<font color="Maroon" title="c2">R</font>,<font color="Olive" title="b1">BASSIGN</font>)</span>)</span><br/>  for <font color="Olive" title="b3">s0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b3">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b2">g</font> &amp; (  for <font color="Olive" title="b4">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b4">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b2">g</font> holds <br/><font color="Olive" title="b4">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Olive" title="b2">g</font> ) holds <br/> ex <font color="Olive" title="b4">pai</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b4">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s0</font> &amp; (  for <font color="Olive" title="b5">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b4">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b2">g</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">BASSIGN</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K31" title="MODELC_1:func.31">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K49" title="MODELC_1:func.49">BASSModel</a> (<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font>)</span>)</span><br/>  for <font color="Olive" title="b2">s0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b2">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b1">g</font> &amp; (  for <font color="Olive" title="b3">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b1">g</font> holds <br/><font color="Olive" title="b3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Olive" title="b1">g</font> ) holds <br/> ex <font color="Olive" title="b3">pai</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b3">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s0</font> &amp; (  for <font color="Olive" title="b4">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b3">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b1">g</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">g</font> be   <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K49" title="MODELC_1:func.49">BASSModel</a> (<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> &amp; (  for <font color="Olive" title="b2">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b2">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> holds <br/><font color="Olive" title="b2">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">g</font> ) holds <br/> ex <font color="Olive" title="b2">pai</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s0</font> &amp; (  for <font color="Olive" title="b3">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b2">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">s0</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> &amp; (  for <font color="Olive" title="b1">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> holds <br/><font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">g</font> ) implies  ex <font color="Olive" title="b1">pai</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font> &amp; (  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> ) ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:168"/><span class="lab"><font color="Green" title="E122">A1</font></span>: 
<font color="Maroon" title="c5">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> &amp;  not <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">g</font> ) or  ex <font color="Olive" title="b1">pai</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font> &amp; (  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> ) ) )</span><br/>

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> </span> } </span> ;<br/>
<span class="kw">assume </span><a NAME="E2:168"/><span class="lab"><font color="Green" title="E123">A2</font></span>: 
 for <font color="Olive" title="b1">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> holds <br/><font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">pai</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font> &amp; (  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> ) )</span><br/>

<a NAME="E3:168"/>
 for <font color="Olive" title="b1">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T1" title="ORDERS_1:th.1">ORDERS_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">Choice</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E5:168"/><span class="lab"><font color="Green" title="E124">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">and </span><br/><a NAME="E6:168"/><span class="lab"><font color="Green" title="E125">A4</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T111" title="FUNCT_1:th.111">FUNCT_1:111</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="modelc_1.html#K3" title="MODELC_1:func.3">UnivF</a> (<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>($1) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">Choice</font>,<font color="Maroon" title="c5">s0</font>);<br/>
<a NAME="E7:168"/><span class="lab"><font color="Green" title="E126">A5</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="168_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" 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="c1">S</font> implies <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:168_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">Y</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="168_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:168_1_1"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font> or  not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="168_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:168_1_1_1"/><span class="lab"><font color="Green" title="E127">A6</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font></span><br/><div class="add"><a NAME="E2:168_1_1_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D3" target="_self" title="MODELC_1:def.3">Def3</a></span>;<br/><a NAME="E3:168_1_1_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:168"><span class="lab"><font color="Green" title="E124">A3</font></span></a>, <a class="txt" href="#E6:168"><span class="lab"><font color="Green" title="E125">A4</font></span></a>, <a class="txt" href="#E1:168_1_1_1"><span class="lab"><font color="Green" title="E127">A6</font></span></a></span>;<br/><a NAME="E4:168_1_1_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:168_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 ;<br/><span class="kw">hence </span><a NAME="E6:168_1_1_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 ; <a class="txt" 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" onclick="hsNdiv(this)" href="javascript:()" title="168_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:168_1_1_2"/>
 not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font></span><br/><div class="add"><a NAME="E2:168_1_1_2"/><span class="kw">then </span>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D3" target="_self" title="MODELC_1:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:168_1_1_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 ; <a class="txt" 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>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c7">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E9:168"/><span class="lab"><font color="Green" title="E127">A7</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Maroon" title="c7">h</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">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E7:168"><span class="lab"><font color="Green" title="E126">A5</font></span></a>);<br/></span>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> $1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>;<br/>
<a NAME="E10:168"/><span class="lab"><font color="Green" title="E128">A8</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">pai</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E12:168"/><span class="lab"><font color="Green" title="E129">A9</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c8">pai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E10:168"><span class="lab"><font color="Green" title="E128">A8</font></span></a>);<br/></span>
<span class="lab"><font color="Green" title="E130">A10</font></span>: <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> = 
<font color="Maroon">H<sub>3</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> )
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:168"><span class="lab"><font color="Green" title="E129">A9</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T84" title="FUNCT_7:th.84">FUNCT_7:84</a></span>
<br/>.= 
<font color="Maroon" title="c5">s0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>
;<br/>
<a NAME="E14:168"/><span class="lab"><font color="Green" title="E131">A11</font></span>: 
<font color="Maroon" title="c5">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 
;<br/>
<a NAME="E15:168"/><span class="lab"><font color="Green" title="E132">A12</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="168_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> );<br/>
<a NAME="E1:168_3"/><span class="lab"><font color="Green" title="E133">A13</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="168_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" 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="c9">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">k1</font> = <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">k2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">p0</font> = <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">p1</font> = <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">p2</font> = <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">Y1</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">Y2</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>;<br/>
<a NAME="E1:168_3_1"/><span class="lab"><font color="Green" title="E134">A14</font></span>: 
<font color="Maroon" title="c5">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:168"><span class="lab"><font color="Green" title="E131">A11</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:168_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>]
 ; <a class="txt" 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="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E3:168_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:168"><span class="lab"><font color="Green" title="E123">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">pai01</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font><span class="kw"> such that </span><br/><a NAME="E5:168_3_1"/><span class="lab"><font color="Green" title="E135">A15</font></span>: 
<font color="Maroon" title="c17">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">and </span><br/><a NAME="E6:168_3_1"/><span class="lab"><font color="Green" title="E136">A16</font></span>: 
<font color="Maroon" title="c17">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T14" target="_self" title="MODELC_1:th.14">Th14</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">x1</font> = <font color="Maroon" title="c17">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1;<br/>
<a NAME="E7:168_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D39" target="_self" title="MODELC_1:def.39">Def41</a></span>;<br/>
<a NAME="E8:168_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A17</font></span>: 
<font color="Maroon" title="c17">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:168_3_1"><span class="lab"><font color="Green" title="E135">A15</font></span></a></span>;<br/>
<a NAME="E9:168_3_1"/>
<font color="Maroon" title="c17">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:168_3_1"><span class="lab"><font color="Green" title="E136">A16</font></span></a></span>;<br/>
<a NAME="E10:168_3_1"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:168_3_1"><span class="lab"><font color="Green" title="E137">A17</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E11:168_3_1"/><span class="kw">then </span>
 not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E12:168_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A18</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E13:168_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A19</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#D2" title="ORDERS_1:def.2">ORDERS_1:def 2</a></span>;<br/>
<a NAME="E14:168_3_1"/><span class="lab"><font color="Green" title="E140">A20</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:168"><span class="lab"><font color="Green" title="E124">A3</font></span></a>, <a class="txt" href="#E12:168_3_1"><span class="lab"><font color="Green" title="E138">A18</font></span></a>, <a class="ref" href="orders_1.html#D2" title="ORDERS_1:def.2">ORDERS_1:def 2</a></span>;<br/>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:168"><span class="lab"><font color="Green" title="E129">A9</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></span>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:168_3_1"><span class="lab"><font color="Green" title="E134">A14</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c9">k</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:168"><span class="lab"><font color="Green" title="E129">A9</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:168"><span class="lab"><font color="Green" title="E127">A7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:168_3_1"><span class="lab"><font color="Green" title="E140">A20</font></span></a>, <a class="ref" href="modelc_1.html#D3" target="_self" title="MODELC_1:def.3">Def3</a></span>
;<br/>
<a NAME="E16:168_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:168"><span class="lab"><font color="Green" title="E125">A4</font></span></a>, <a class="txt" href="#E13:168_3_1"><span class="lab"><font color="Green" title="E139">A19</font></span></a></span>;<br/>
<a NAME="E17:168_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E18:168_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A21</font></span>: 
 ex <font color="Olive" title="b1">q1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b1">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">q1</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> )
 
;<br/>
<a NAME="E19:168_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:168"><span class="lab"><font color="Green" title="E123">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">pai02</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font><span class="kw"> such that </span><br/><a NAME="E21:168_3_1"/><span class="lab"><font color="Green" title="E142">A22</font></span>: 
<font color="Maroon" title="c19">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E22:168_3_1"/><span class="lab"><font color="Green" title="E143">A23</font></span>: 
<font color="Maroon" title="c19">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T14" target="_self" title="MODELC_1:th.14">Th14</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">x2</font> = <font color="Maroon" title="c19">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1;<br/>
<a NAME="E23:168_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D39" target="_self" title="MODELC_1:def.39">Def41</a></span>;<br/>
<a NAME="E24:168_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">A24</font></span>: 
<font color="Maroon" title="c19">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:168_3_1"><span class="lab"><font color="Green" title="E142">A22</font></span></a></span>;<br/>
<a NAME="E25:168_3_1"/>
<font color="Maroon" title="c19">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:168_3_1"><span class="lab"><font color="Green" title="E143">A23</font></span></a></span>;<br/>
<a NAME="E26:168_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:168_3_1"><span class="lab"><font color="Green" title="E144">A24</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E27:168_3_1"/><span class="kw">then </span>
 not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E28:168_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E145">A25</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E29:168_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E146">A26</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#D2" title="ORDERS_1:def.2">ORDERS_1:def 2</a></span>;<br/>
<a NAME="E30:168_3_1"/><span class="lab"><font color="Green" title="E147">A27</font></span>: 
<font color="Maroon" title="c5">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:168"><span class="lab"><font color="Green" title="E131">A11</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E31:168_3_1"/><span class="lab"><font color="Green" title="E148">A28</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:168"><span class="lab"><font color="Green" title="E124">A3</font></span></a>, <a class="txt" href="#E28:168_3_1"><span class="lab"><font color="Green" title="E145">A25</font></span></a>, <a class="ref" href="orders_1.html#D2" title="ORDERS_1:def.2">ORDERS_1:def 2</a></span>;<br/>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon">H<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:168"><span class="lab"><font color="Green" title="E129">A9</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T71" title="FUNCT_7:th.71">FUNCT_7:71</a></span>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:168_3_1"><span class="lab"><font color="Green" title="E147">A27</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:168"><span class="lab"><font color="Green" title="E129">A9</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:168"><span class="lab"><font color="Green" title="E127">A7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:168_3_1"><span class="lab"><font color="Green" title="E148">A28</font></span></a>, <a class="ref" href="modelc_1.html#D3" target="_self" title="MODELC_1:def.3">Def3</a></span>
;<br/>
<a NAME="E33:168_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:168"><span class="lab"><font color="Green" title="E125">A4</font></span></a>, <a class="txt" href="#E29:168_3_1"><span class="lab"><font color="Green" title="E146">A26</font></span></a></span>;<br/>
<a NAME="E34:168_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E35:168_3_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">q2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b1">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b1">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E36:168_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:168_3_1"><span class="lab"><font color="Green" title="E141">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:168_3"/><span class="lab"><font color="Green" title="E134">A29</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="168_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c9">Y</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:168_3_2"/>
<font color="Maroon" title="c5">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K26" title="MODELC_1:func.26">EX</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:168"><span class="lab"><font color="Green" title="E122">A1</font></span></a>, <a class="txt" href="#E2:168"><span class="lab"><font color="Green" title="E123">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">pai01</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font><span class="kw"> such that </span><br/><a NAME="E3:168_3_2"/><span class="lab"><font color="Green" title="E135">A30</font></span>: 
<font color="Maroon" title="c11">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font>
 <span class="kw">and </span><br/><a NAME="E4:168_3_2"/><span class="lab"><font color="Green" title="E136">A31</font></span>: 
<font color="Maroon" title="c11">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T14" target="_self" title="MODELC_1:th.14">Th14</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c11">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1;<br/>
<a NAME="E5:168_3_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D39" target="_self" title="MODELC_1:def.39">Def41</a></span>;<br/>
<a NAME="E6:168_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A32</font></span>: 
<font color="Maroon" title="c11">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:168_3_2"><span class="lab"><font color="Green" title="E135">A30</font></span></a></span>;<br/>
<a NAME="E7:168_3_2"/>
<font color="Maroon" title="c11">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:168_3_2"><span class="lab"><font color="Green" title="E136">A31</font></span></a></span>;<br/>
<a NAME="E8:168_3_2"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:168_3_2"><span class="lab"><font color="Green" title="E137">A32</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E9:168_3_2"/><span class="kw">then </span>
 not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E10:168_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A33</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E11:168_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A34</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:168"><span class="lab"><font color="Green" title="E124">A3</font></span></a>, <a class="ref" href="orders_1.html#D2" title="ORDERS_1:def.2">ORDERS_1:def 2</a></span>;<br/>
<a NAME="E12:168_3_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:168_3_2"><span class="lab"><font color="Green" title="E138">A33</font></span></a>, <a class="ref" href="orders_1.html#D2" title="ORDERS_1:def.2">ORDERS_1:def 2</a></span>;<br/>
<a NAME="E13:168_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A35</font></span>: 
<font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:168"><span class="lab"><font color="Green" title="E125">A4</font></span></a></span>;<br/>
<a NAME="E14:168_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E16:168_3_2"/><span class="lab"><font color="Green" title="E141">A36</font></span>: 
<font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t</font>
 <span class="kw">and </span><br/><a NAME="E17:168_3_2"/><span class="lab"><font color="Green" title="E142">A37</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">s0</font>,<font color="Maroon" title="c13">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 ;<br/>
<a NAME="E18:168_3_2"/>
<font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:168_3_2"><span class="lab"><font color="Green" title="E140">A35</font></span></a>, <a class="txt" href="#E16:168_3_2"><span class="lab"><font color="Green" title="E141">A36</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">s1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E20:168_3_2"/><span class="lab"><font color="Green" title="E143">A38</font></span>: 
<font color="Maroon" title="c13">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s1</font>
 <span class="kw">and </span><br/><a NAME="E21:168_3_2"/>
<font color="Maroon" title="c14">s1</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font>
 ;<br/>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 = 
<font color="Maroon">H<sub>3</sub></font>(1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:168"><span class="lab"><font color="Green" title="E129">A9</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">|**</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T70" title="FUNCT_7:th.70">FUNCT_7:70</a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">s0</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:168"><span class="lab"><font color="Green" title="E127">A7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c14">s1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:168_3_2"><span class="lab"><font color="Green" title="E139">A34</font></span></a>, <a class="txt" href="#E16:168_3_2"><span class="lab"><font color="Green" title="E141">A36</font></span></a>, <a class="txt" href="#E20:168_3_2"><span class="lab"><font color="Green" title="E143">A38</font></span></a>, <a class="ref" href="modelc_1.html#D3" target="_self" title="MODELC_1:def.3">Def3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E23:168_3_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:168"><span class="lab"><font color="Green" title="E122">A1</font></span></a>, <a class="txt" href="#E13:168"><span class="lab"><font color="Green" title="E130">A10</font></span></a>, <a class="txt" href="#E17:168_3_2"><span class="lab"><font color="Green" title="E142">A37</font></span></a>, <a class="txt" href="#E20:168_3_2"><span class="lab"><font color="Green" title="E143">A38</font></span></a></span>; <a class="txt" 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="E3:168_3"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:168_3"><span class="lab"><font color="Green" title="E134">A29</font></span></a>, <a class="txt" href="#E1:168_3"><span class="lab"><font color="Green" title="E133">A13</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:168_3"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> )
 ; <a class="txt" 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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">pai</font> = <font color="Maroon" title="c8">pai</font> as    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D39" target="_self" title="MODELC_1:def.39">Def41</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">pai</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font> &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E17:168"/>
( <font color="Maroon" title="c9">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font> &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:168"><span class="lab"><font color="Green" title="E130">A10</font></span></a>, <a class="txt" href="#E15:168"><span class="lab"><font color="Green" title="E132">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
