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

<b>set </b><font color="Maroon" title="c3">D</font> = <span class="p1">{<span class="default"> <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>,<font color="Olive" title="b1">f</font> #) where <font color="Olive" title="b1">f</font> is    <a href="grcat_1.html#M6" title="GRCAT_1:mode.6">Element</a> of  <a href="grcat_1.html#K17" title="GRCAT_1:func.17">Maps</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> : <font color="Olive" title="b1">f</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> </span>}</span> ;<br/>
<a NAME="E1:72_1_1"/>
 <a href="grcat_1.html#K8" title="GRCAT_1:func.8">ZeroMap</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">f0</font> =  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">ZeroMap</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> as    <a href="grcat_1.html#M6" title="GRCAT_1:mode.6">Element</a> of  <a href="grcat_1.html#K17" title="GRCAT_1:func.17">Maps</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> ;<br/>
<a NAME="E3:72_1_1"/>
<font color="Maroon" title="c4">f0</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T16" target="_self" title="GRCAT_1:th.16">Th16</a></i>;<br/>
<a NAME="E4:72_1_1"/><b>then </b>
 <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c4">f0</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>,<font color="Olive" title="b1">f</font> #) where <font color="Olive" title="b1">f</font> is    <a href="grcat_1.html#M6" title="GRCAT_1:mode.6">Element</a> of  <a href="grcat_1.html#K17" title="GRCAT_1:func.17">Maps</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> : <font color="Olive" title="b1">f</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">D</font> = <span class="p1">{<span class="default"> <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>,<font color="Olive" title="b1">f</font> #) where <font color="Olive" title="b1">f</font> is    <a href="grcat_1.html#M6" title="GRCAT_1:mode.6">Element</a> of  <a href="grcat_1.html#K17" title="GRCAT_1:func.17">Maps</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> : <font color="Olive" title="b1">f</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E6:72_1_1"/><i><font color="Green" title="E39">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> holds <br/><font color="Olive" title="b1">x</font> is  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:72_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 ;<br/>

<a NAME="E2:72_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">f</font> being   <a href="grcat_1.html#M6" title="GRCAT_1:mode.6">Element</a> of  <a href="grcat_1.html#K17" title="GRCAT_1:func.17">Maps</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>,<font color="Olive" title="b1">f</font> #) &amp; <font color="Olive" title="b1">f</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> )
 
;<br/>
<b>hence </b><a NAME="E3:72_1_1_1"/>
<font color="Maroon" title="c6">x</font> is  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="ref" href="grcat_1.html#T20" target="_self" title="GRCAT_1:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:72_1_1"/><b>then </b><i><font color="Green" title="E40">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">D</font> holds  <font color="Olive" title="b1">x</font> is  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>
 
;<br/>
<a NAME="E8:72_1_1"/><i><font color="Green" title="E41">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> is  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</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">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:72_1_1_2"/>
<font color="Maroon" title="c6">x</font> is  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> ;<br/>
<a NAME="E3:72_1_1_2"/>
(  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">dom</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="grcat_1.html#K10" title="GRCAT_1:func.10">cod</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> )
 
<b>by </b><i><a class="ref" href="grcat_1.html#D19" target="_self" title="GRCAT_1:def.19">Def19</a></i>;<br/>
<a NAME="E4:72_1_1_2"/><b>then </b><i><font color="Green" title="E42">A4</font></i>: 
( the <a href="grcat_1.html#U1" title="GRCAT_1:sel.1">Dom</a> of <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp; the <a href="grcat_1.html#U2" title="GRCAT_1:sel.2">Cod</a> of <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> )
 
;<br/>
<a NAME="E5:72_1_1_2"/><i><font color="Green" title="E43">A5</font></i>: 
the <a href="grcat_1.html#U3" title="GRCAT_1:sel.3">Fun</a> of <font color="Maroon" title="c7">x</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T18" target="_self" title="GRCAT_1:th.18">Th18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">f</font> = the <a href="grcat_1.html#U3" title="GRCAT_1:sel.3">Fun</a> of <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> <b>by </b><i><a class="txt" href="grcat_1.html#E4:72_1_1_2"><i><font color="Green" title="E42">A4</font></i></a></i>;<br/>
<a NAME="E7:72_1_1_2"/>
<font color="Maroon" title="c8">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">f</font> as    <a href="grcat_1.html#M6" title="GRCAT_1:mode.6">Element</a> of  <a href="grcat_1.html#K17" title="GRCAT_1:func.17">Maps</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> ;<br/>
<a NAME="E9:72_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c9">g</font> #)
 
<b>by </b><i><a class="txt" href="grcat_1.html#E4:72_1_1_2"><i><font color="Green" title="E42">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:72_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="grcat_1.html#E5:72_1_1_2"><i><font color="Green" title="E43">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c6">D</font> = <font color="Maroon" title="c5">D</font> as    <a href="grcat_1.html#M4" title="GRCAT_1:mode.4">GroupMorphism_DOMAIN</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> <b>by </b><i><a class="txt" href="grcat_1.html#E7:72_1_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="grcat_1.html#T38" target="_self" title="GRCAT_1:th.38">Th38</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">D</font>
;<br/>

<b>thus </b><a NAME="E10:72_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font> iff <font color="Olive" title="b1">x</font> is  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> )
 <b>by </b><i><a class="txt" href="grcat_1.html#E6:72_1_1"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="grcat_1.html#E8:72_1_1"><i><font color="Green" title="E41">A3</font></i></a></i>;<br/>


</div>
