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

<span class="kw">let </span><font color="Maroon" title="c1">UN</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span> holds <br/> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; ( for <font color="Olive" title="b2">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>  st  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) &amp; ( for <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">C</font> =  <a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">V</font> =  <a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">X</font> =  <a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; ( for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>  st  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> ) &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> holds <br/><font color="Olive" title="b1">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">b9</font> = <font color="Maroon" title="c5">b</font> as    <a href="ringcat1.html#M2" title="RINGCAT1:mode.2">Element</a> of  <a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">b99</font> = <font color="Maroon" title="c6">b9</font> as   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> ;<br/>
<a NAME="E3:82"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ringcat1.html#K13" title="RINGCAT1:func.13">ID</a> <font color="Maroon" title="c6">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ringcat1.html#T28" target="_self" title="RINGCAT1:th.28">Th28</a></span>;<br/>
<span class="kw">hence </span><span class="lab"><font color="Green" title="E52">A2</font></span>:  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> = 
 <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K5" title="RINGCAT1:func.5">ID</a> <font color="Maroon" title="c7">b99</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ringcat1.html#D20" target="_self" title="RINGCAT1:def.20">Def20</a></span>
<br/>.= 
<font color="Maroon" title="c5">b</font>

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; ( for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>  st  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> ) &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> holds <br/><font color="Olive" title="b1">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) )</span><br/>

<span class="kw">thus </span><span class="lab"><font color="Green" title="E53">A3</font></span>:  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> = 
 <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K5" title="RINGCAT1:func.5">ID</a> <font color="Maroon" title="c7">b99</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="ringcat1.html#D21" target="_self" title="RINGCAT1:def.21">Def21</a></span>
<br/>.= 
<font color="Maroon" title="c5">b</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>  st  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> ) &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> holds <br/><font color="Olive" title="b1">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E6:82"/>
for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>  st  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> holds <br/><font color="Olive" title="b1">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> implies <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:82_3"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">f1</font> = <font color="Maroon" title="c8">f</font> as   <a href="ringcat1.html#V2" title="RINGCAT1:attr.2">strict</a>   <a href="ringcat1.html#M3" title="RINGCAT1:mode.3">Element</a> of  <a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="ringcat1.html#T27" target="_self" title="RINGCAT1:th.27">Th27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">f9</font> = <font color="Maroon" title="c9">f1</font> as   <a href="ringcat1.html#V2" title="RINGCAT1:attr.2">strict</a>  <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a> ;<br/>
<a NAME="E4:82_3"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
 <a href="ringcat1.html#K2" title="RINGCAT1:func.2">cod</a> <font color="Maroon" title="c10">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_3"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="ringcat1.html#D21" target="_self" title="RINGCAT1:def.21">Def21</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c8">f</font> = 
<span class="p1">(<span class="default"><a href="ringcat1.html#K5" title="RINGCAT1:func.5">ID</a> <font color="Maroon" title="c7">b99</font></span>)</span> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <font color="Maroon" title="c10">f9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E4:82"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E1:82_3"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="ringcat1.html#T30" target="_self" title="RINGCAT1:th.30">Th30</a></span>
<br/>.= 
<font color="Maroon" title="c8">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_3"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="ringcat1.html#T14" target="_self" title="RINGCAT1:th.14">Th14</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E7:82"/>
for <font color="Olive" title="b1">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>  st  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> holds <br/><font color="Olive" title="b1">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ringcat1.html#K18" title="RINGCAT1:func.18">RingCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> implies <font color="Maroon" title="c8">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:82_4"/><span class="lab"><font color="Green" title="E54">A6</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">f1</font> = <font color="Maroon" title="c8">f</font> as   <a href="ringcat1.html#V2" title="RINGCAT1:attr.2">strict</a>   <a href="ringcat1.html#M3" title="RINGCAT1:mode.3">Element</a> of  <a href="ringcat1.html#K10" title="RINGCAT1:func.10">Morphs</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K9" title="RINGCAT1:func.9">RingObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="ringcat1.html#T27" target="_self" title="RINGCAT1:th.27">Th27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">f9</font> = <font color="Maroon" title="c9">f1</font> as   <a href="ringcat1.html#V2" title="RINGCAT1:attr.2">strict</a>  <a href="ringcat1.html#NM1" title="RINGCAT1:NM.1">RingMorphism</a> ;<br/>
<a NAME="E4:82_4"/><span class="lab"><font color="Green" title="E55">A7</font></span>: 
 <a href="ringcat1.html#K1" title="RINGCAT1:func.1">dom</a> <font color="Maroon" title="c10">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_4"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="ringcat1.html#D20" target="_self" title="RINGCAT1:def.20">Def20</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c8">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <font color="Maroon" title="c5">b</font></span>)</span> = 
<font color="Maroon" title="c10">f9</font> <a href="ringcat1.html#K6" title="RINGCAT1:func.6">*</a> <span class="p1">(<span class="default"><a href="ringcat1.html#K5" title="RINGCAT1:func.5">ID</a> <font color="Maroon" title="c7">b99</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E5:82"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E1:82_4"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="ringcat1.html#T30" target="_self" title="RINGCAT1:th.30">Th30</a></span>
<br/>.= 
<font color="Maroon" title="c8">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_4"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="ref" href="ringcat1.html#T14" target="_self" title="RINGCAT1:th.14">Th14</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
