<?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="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <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">h</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><br/>  for <font color="Olive" title="b2">A</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Olive" title="b1">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Olive" title="b2">A</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Olive" title="b1">h</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">h</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">  for <font color="Olive" title="b1">A</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Olive" title="b1">A</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">h</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</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="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c4">A</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">h</font></span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c5">C</font> = <font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c4">A</font> as   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">h</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="grsolv_1.html#T9" onmouseover="rs('grsolv_1/T9')" onmouseout="rh()">GRSOLV_1:9</a></span>;<br/>
<a NAME="E2:37"/>
 for <font color="Olive" title="b1">b</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">h</font></span>)</span> holds  <font color="Olive" title="b1">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">b</font> be   <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">h</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="c6">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c6">b</font></span><br/>

<a NAME="E1:37_1"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c6">b</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">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1"><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> <font color="Maroon" title="c6">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">C</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c6">b</font></span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c7">b1</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="E2:37_1_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="group_6.html#T45" onmouseover="rs('group_6/T45')" onmouseout="rh()">GROUP_6:45</a></span>;<br/><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> <font color="Maroon" title="c6">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">C</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c6">b</font> )</span><br/><span class="kw">assume </span><a NAME="E3:37_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">C</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> <font color="Maroon" title="c5">C</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c6">b</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">g</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">h</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:37_1_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">g</font>
 <span class="kw">and </span><br/><a NAME="E6:37_1_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T103" onmouseover="rs('group_2/T103')" onmouseout="rh()">GROUP_2:103</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">g1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">A</font><span class="kw"> such that </span><br/><a NAME="E8:37_1_1"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K1" title="GRSOLV_1:func.1">|</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="group_6.html#T45" onmouseover="rs('group_6/T45')" onmouseout="rh()">GROUP_6:45</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">g1</font> = <font color="Maroon" title="c10">g1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T42" onmouseover="rs('group_2/T42')" onmouseout="rh()">GROUP_2:42</a></span>;<br/><a NAME="E10:37_1_1"/>
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A6</font></span>: <font color="Maroon" title="c8">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b1</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">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E29">A3</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">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">g1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" onmouseover="rs('group_6/D6')" onmouseout="rh()">GROUP_6:def 6</a></span>
;<br/><a NAME="E12:37_1_1"/>
<font color="Maroon" title="c11">g1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/><a NAME="E13:37_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<font color="Maroon" title="c7">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">b1</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T103" onmouseover="rs('group_2/T103')" onmouseout="rh()">GROUP_2:103</a></span>;<br/><a NAME="E14:37_1_1"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
<font color="Maroon" title="c3">h</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c7">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grsolv_1.html#T13" onmouseover="rs('grsolv_1/T13')" onmouseout="rh()">GRSOLV_1:13</a></span>;<br/><a NAME="E15:37_1_1"/>
<font color="Maroon" title="c7">b1</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c7">b1</font>
 <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/><a NAME="E16:37_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E13:37_1_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E14:37_1_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</a></span>;<br/><span class="kw">hence </span><a NAME="E17:37_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="group_6.html#T2" onmouseover="rs('group_6/T2')" onmouseout="rh()">GROUP_6:2</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="E3:37_1"/>
<font color="Maroon" title="c6">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c6">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">hence </span><a NAME="E3:37"/>
<font color="Maroon" title="c3">h</font> <a href="grsolv_1.html#K2" title="GRSOLV_1:func.2">.:</a> <font color="Maroon" title="c4">A</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_6.html#K10" title="GROUP_6:func.10">Image</a> <font color="Maroon" title="c3">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T118" onmouseover="rs('group_3/T118')" onmouseout="rh()">GROUP_3:118</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>
