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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="vectsp_1.html#NM1" title="VECTSP_1:NM.1">AddGroup</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">f</font> being   <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>  st  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> holds <br/><span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</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="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>  st  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> holds <br/><font color="Olive" title="b1">g</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</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">set </span><font color="Maroon" title="c2">i</font> =  <a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font>;<br/>
<span class="kw">thus </span><a NAME="E1:48"/>
 <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">f</font> being   <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>  st  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> holds <br/><span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</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="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>  st  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> holds <br/><font color="Olive" title="b1">g</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</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="E2:48"/>
 <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</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">f</font> being   <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>  st  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> holds <br/><span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</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="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>  st  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> holds <br/><font color="Olive" title="b1">g</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</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="E3:48"/>
 for <font color="Olive" title="b1">f</font> being   <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>  st  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> holds <br/><span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>  st  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> holds <br/><font color="Olive" title="b1">g</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</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="48_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> implies <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">H</font> =  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Maroon" title="c3">f</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">f9</font> = <font color="Maroon" title="c3">f</font> as    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="grcat_1.html#D14" target="_self" title="GRCAT_1:def.14">Def19</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">m</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E4:48_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c5">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <span class="p1">(<span class="default"><a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">m</font> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#T13" target="_self" title="GRCAT_1:th.13">Th23</a></span>;<br/>
<a NAME="E5:48_1"/>
(  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">m</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:48_1"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E4:48_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="grcat_1.html#D16" target="_self" title="GRCAT_1:def.16">Def21</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>

<span class="kw">thus </span><a NAME="E4:48"/>
 for <font color="Olive" title="b1">g</font> being   <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>  st  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> holds <br/><font color="Olive" title="b1">g</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> implies <font color="Maroon" title="c3">f</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_2"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</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="c3">f</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">H</font> =  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Maroon" title="c3">f</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">f9</font> = <font color="Maroon" title="c3">f</font> as    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>, <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Maroon" title="c3">f</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_2"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="grcat_1.html#D14" target="_self" title="GRCAT_1:def.14">Def19</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">m</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Maroon" title="c3">f</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:48_2"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c5">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c6">m</font> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#T13" target="_self" title="GRCAT_1:th.13">Th23</a></span>;<br/>
<a NAME="E5:48_2"/>
(  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c6">m</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:48_2"/>
<font color="Maroon" title="c3">f</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_2"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E4:48_2"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="grcat_1.html#D16" target="_self" title="GRCAT_1:def.16">Def21</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>


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