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

<b>let </b><font color="Maroon" title="c1">h</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">h</font> &amp; ( 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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">h</font> holds <br/><span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <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> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> ) )</span><br/>

<b>thus </b><i><font color="Green" title="E41">A1</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> = 
 <a href="funct_6.html#K4" title="FUNCT_6:func.4">meet</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_6.html#T49" target="_self" title="FUNCT_6:th.49">Th49</a></i>
<br/>.= 
 <a href="funct_6.html#K4" title="FUNCT_6:func.4">meet</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">h</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="funct_6.html#T33" target="_self" title="FUNCT_6:th.33">Th33</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">h</font>
<b>by </b><i><a class="ref" href="funct_6.html#T39" target="_self" title="FUNCT_6:th.39">Th39</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">h</font> holds <br/><span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <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> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><br/>

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</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> <font color="Maroon" title="c1">h</font> implies <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )</span><br/>

<b>assume </b><a NAME="E2:67"/><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c2">x</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> <font color="Maroon" title="c1">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><br/>

<a NAME="E3:67"/><b>then </b>
( <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="funct_6.html#E1:67"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="funct_6.html#T49" target="_self" title="FUNCT_6:th.49">Th49</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:67"/><b>then </b>
( <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> &amp;  <a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">h</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="funct_6.html#T33" target="_self" title="FUNCT_6:th.33">Th33</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:67"/><i><font color="Green" title="E43">A3</font></i>: 
( <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">h</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">h</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">h</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E7:67"/><i><font color="Green" title="E44">A4</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> )
 
<b>by </b><i><a class="txt" href="funct_6.html#E6:67"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">g</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E9:67"/>
( <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c2">x</font> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="funct_6.html#E1:67"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="funct_6.html#E2:67"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="funct_6.html#E6:67"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="funct_6.html#E7:67"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="funct_6.html#T51" target="_self" title="FUNCT_6:th.51">Th51</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="funct_5.html#T45" title="FUNCT_5:th.45">FUNCT_5:45</a></i>;<br/>
<b>hence </b><a NAME="E10:67"/>
<span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">h</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="funct_6.html#E6:67"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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