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

<span class="kw">let </span><font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c2">H</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> holds <br/> ( <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Olive" title="b1">g</font></span>)</span>, <a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Olive" title="b1">g</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a>  &amp;  ex <font color="Olive" title="b2">h</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Olive" title="b1">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Olive" title="b1">g</font></span>)</span> st <br/>( <font color="Olive" title="b2">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Olive" title="b1">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Olive" title="b1">g</font></span>)</span></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">g</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>, <a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a>  &amp;  ex <font color="Olive" title="b1">h</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font></span>)</span> st <br/>( <font color="Olive" title="b1">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">I</font> = <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">J</font> =  <a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>;<br/>
<a NAME="E1:99"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:99"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
 for <font color="Olive" title="b1">S</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>  ex <font color="Olive" title="b2">T</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">T</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">S</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">T</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">S</font>,<font color="Olive" title="b1">T</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E2:99_1"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
<font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:99_1"/>
<font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T13" target="_self" onmouseover="rs('group_6/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<a NAME="E4:99_1"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T45" target="_self" onmouseover="rs('group_6/T45')" onmouseout="rh()">Th45</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">T</font> = <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font></span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">T</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c8">T</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> implies <font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E6:99_1"/>
<font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span><br/>

<a NAME="E7:99_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_1"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="ref" href="group_2.html#T114" onmouseover="rs('group_2/T114')" onmouseout="rh()">GROUP_2:114</a></span>;<br/>
<span class="kw">then </span> <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">H</font> = 
<font color="Maroon" title="c3">g</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">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T41" target="_self" onmouseover="rs('group_6/T41')" onmouseout="rh()">Th41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" target="_self" onmouseover="rs('group_6/D6')" onmouseout="rh()">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T32" target="_self" onmouseover="rs('group_6/T32')" onmouseout="rh()">Th32</a></span>
;<br/>
<a NAME="E9:99_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T12" onmouseover="rs('group_1/T12')" onmouseout="rh()">GROUP_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:99_1"/>
<font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:99"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
 for <font color="Olive" title="b1">S</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E58">A2</font></span></a>);<br/></span>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font>, <font color="Olive" title="b2">T</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> holds  <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">T</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">S</font>, <font color="Maroon" title="c8">T</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">S</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">T</font></span>)</span></span><br/><span class="kw">consider </span><font color="Maroon" title="c9">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E2:99_2"/><span class="lab"><font color="Green" title="E60">A5</font></span>: 
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:99_2"/>
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T13" target="_self" onmouseover="rs('group_6/T13')" onmouseout="rh()">Th13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E5:99_2"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
<font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:99_2"/><span class="lab"><font color="Green" title="E62">A7</font></span>: 
<font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T13" target="_self" onmouseover="rs('group_6/T13')" onmouseout="rh()">Th13</a></span>;<br/><a NAME="E7:99_2"/><span class="lab"><font color="Green" title="E63">A8</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E5:99_2"><span class="lab"><font color="Green" title="E61">A6</font></span></a></span>;<br/><a NAME="E8:99_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E2:99_2"><span class="lab"><font color="Green" title="E60">A5</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A9</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">S</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">T</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99_2"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="ref" href="group_2.html#T43" onmouseover="rs('group_2/T43')" onmouseout="rh()">GROUP_2:43</a></span>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" target="_self" onmouseover="rs('group_6/D6')" onmouseout="rh()">Def6</a></span>
;<br/><font color="Maroon" title="c7">S</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">T</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_2"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="txt" href="#E6:99_2"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="ref" href="group_6.html#D3" target="_self" onmouseover="rs('group_6/D3')" onmouseout="rh()">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p3">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T11" onmouseover="rs('group_3/T11')" onmouseout="rh()">GROUP_3:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c10">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T5" target="_self" onmouseover="rs('group_6/T5')" onmouseout="rh()">Th5</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c10">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T106" onmouseover="rs('group_2/T106')" onmouseout="rh()">GROUP_2:106</a></span>
<br/>.= 
<font color="Maroon" title="c9">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T117" onmouseover="rs('group_3/T117')" onmouseout="rh()">GROUP_3:117</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T105" onmouseover="rs('group_2/T105')" onmouseout="rh()">GROUP_2:105</a></span>
;<br/><span class="kw">hence </span><a NAME="E11:99_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">S</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">T</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E9:99_2"><span class="lab"><font color="Green" title="E64">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" target="_self" onmouseover="rs('group_6/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E7:99"/><span class="lab"><font color="Green" title="E60">A10</font></span>: 
<font color="Maroon" title="c7">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">y1</font>, <font color="Maroon" title="c9">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">f</font> or  not <font color="Maroon" title="c9">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">f</font> or  not <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y2</font> or <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:99_3"/>
( <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c9">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y2</font> or <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">S</font> = <font color="Maroon" title="c8">y1</font>, <font color="Maroon" title="c11">T</font> = <font color="Maroon" title="c9">y2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> ;<br/>
<span class="kw">assume </span><a NAME="E3:99_3"/><span class="lab"><font color="Green" title="E61">A11</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E5:99_3"/><span class="lab"><font color="Green" title="E62">A12</font></span>: 
<font color="Maroon" title="c10">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:99_3"/>
<font color="Maroon" title="c10">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T13" target="_self" onmouseover="rs('group_6/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E8:99_3"/><span class="lab"><font color="Green" title="E63">A13</font></span>: 
<font color="Maroon" title="c11">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:99_3"/>
<font color="Maroon" title="c11">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c13">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T13" target="_self" onmouseover="rs('group_6/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<a NAME="E10:99_3"/><span class="lab"><font color="Green" title="E64">A14</font></span>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E8:99_3"><span class="lab"><font color="Green" title="E63">A13</font></span></a></span>;<br/>
<a NAME="E11:99_3"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E5:99_3"><span class="lab"><font color="Green" title="E62">A12</font></span></a></span>;<br/>
<a NAME="E12:99_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_3"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="txt" href="#E10:99_3"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="ref" href="group_1.html#D5" onmouseover="rs('group_1/D5')" onmouseout="rh()">GROUP_1:def 5</a></span>;<br/>
<span class="kw">then </span> <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">H</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T32" target="_self" onmouseover="rs('group_6/T32')" onmouseout="rh()">Th32</a></span>
<br/>.= 
<font color="Maroon" title="c3">g</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="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" target="_self" onmouseover="rs('group_6/D6')" onmouseout="rh()">Def6</a></span>
;<br/>
<a NAME="E14:99_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T41" target="_self" onmouseover="rs('group_6/T41')" onmouseout="rh()">Th41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:99_3"/>
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:99_3"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="txt" href="#E8:99_3"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="group_2.html#T114" onmouseover="rs('group_2/T114')" onmouseout="rh()">GROUP_2:114</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:99"/><span class="lab"><font color="Green" title="E61">A15</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E62">A16</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> implies ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> ) ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> implies ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> ) ) &amp; ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:99_4"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> implies ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> ) )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:99_4_1"/><span class="lab"><font color="Green" title="E63">A17</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> )</span><br/>

<span class="kw">hence </span><a NAME="E2:99_4_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span><br/>

<a NAME="E3:99_4_1"/><span class="lab"><font color="Green" title="E64">A18</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:99_4_1"/>
<span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E1:99_4_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:99_4_1"/>
<span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_4_1"><span class="lab"><font color="Green" title="E64">A18</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:99_4"/><span class="lab"><font color="Green" title="E63">A19</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:99_4"/>
<span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font></span><br/><span class="kw">thus </span><a NAME="E4:99_4"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:99"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="txt" href="#E2:99_4"><span class="lab"><font color="Green" title="E63">A19</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:99"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font></span>)</span> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:99_5"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font></span><br/>

<a NAME="E2:99_5"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E4:99_5"/><span class="lab"><font color="Green" title="E63">A20</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T45" target="_self" onmouseover="rs('group_6/T45')" onmouseout="rh()">Th45</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">S</font> = <font color="Maroon" title="c9">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T14" target="_self" onmouseover="rs('group_6/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E6:99_5"/>
( <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:99_5"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_5"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:99"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A21</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
;<br/>
<a NAME="E12:99"/><span class="kw">then </span>
<font color="Maroon" title="c7">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:99"/>
<font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>, <a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">g</font></span>)</span> st <br/>( <font color="Olive" title="b1">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">f</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Maroon" title="c3">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E14:99"/>
<font color="Maroon" title="c7">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:99"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_6"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> 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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p3">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> implies <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p3">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:99_6"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p3">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E3:99_6"/>
<span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D8" target="_self" onmouseover="rs('group_6/D8')" onmouseout="rh()">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E4:99_6"/>
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p3">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E59">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:99"/>
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K8" title="GROUP_6:func.8">nat_hom</a> <span class="p2">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">Ker</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="ref" href="funct_1.html#T10" onmouseover="rs('funct_1/T10')" onmouseout="rh()">FUNCT_1:10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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