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

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <a href="grcat_1.html#G1" target="_self">GroupMorphismStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> #) where B is    <a href="grcat_1.html#M6" target="_self">Element</a> of  <a href="grcat_1.html#K17" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="grcat_1.html#V1" target="_self">additive</a> </span>}</span> ;<br/>
<a NAME="E1:74_1_1"/>
 <a href="grcat_1.html#K8" target="_self">ZeroMap</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="grcat_1.html#K8" target="_self">ZeroMap</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> as    <a href="grcat_1.html#M6" target="_self">Element</a> of  <a href="grcat_1.html#K17" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E3:74_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="grcat_1.html#V1" target="_self">additive</a>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E4:74_1_1"/><b>then </b>
 <a href="grcat_1.html#G1" target="_self">GroupMorphismStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> #) <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <a href="grcat_1.html#G1" target="_self">GroupMorphismStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> #) where B is    <a href="grcat_1.html#M6" target="_self">Element</a> of  <a href="grcat_1.html#K17" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="grcat_1.html#V1" target="_self">additive</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <a href="grcat_1.html#G1" target="_self">GroupMorphismStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> #) where B is    <a href="grcat_1.html#M6" target="_self">Element</a> of  <a href="grcat_1.html#K17" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="grcat_1.html#V1" target="_self">additive</a> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E6:74_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is  <a href="grcat_1.html#V2" target="_self">strict</a>  <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:74_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:74_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="grcat_1.html#M6" target="_self">Element</a> of  <a href="grcat_1.html#K17" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#G1" target="_self">GroupMorphismStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> #) &amp; <font color="Olive">b<sub>1</sub></font> is <a href="grcat_1.html#V1" target="_self">additive</a> )
 
;<br/>
<b>hence </b><a NAME="E3:74_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is  <a href="grcat_1.html#V2" target="_self">strict</a>  <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="grcat_1.html#T20" target="_self">Th20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:74_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> holds <font color="Olive">b<sub>1</sub></font> is  <a href="grcat_1.html#V2" target="_self">strict</a>  <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E8:74_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> is  <a href="grcat_1.html#V2" target="_self">strict</a>  <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:74_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> is  <a href="grcat_1.html#V2" target="_self">strict</a>  <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  <a href="grcat_1.html#V2" target="_self">strict</a>  <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E3:74_1_1_2"/>
(  <a href="grcat_1.html#K9" target="_self">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="grcat_1.html#K10" target="_self">cod</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="grcat_1.html#D19" target="_self">Def19</a></i>;<br/>
<a NAME="E4:74_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
( the <a href="grcat_1.html#U1" target="_self">Dom</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="grcat_1.html#U2" target="_self">Cod</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<a NAME="E5:74_1_1_2"/><i><font color="Green">E43</font></i>: 
the <a href="grcat_1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>7</sub></font> is <a href="grcat_1.html#V1" target="_self">additive</a>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T18" target="_self">Th18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="grcat_1.html#U3" target="_self">Fun</a> of <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="grcat_1.html#E4:74_1_1_2"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E7:74_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="grcat_1.html#M6" target="_self">Element</a> of  <a href="grcat_1.html#K17" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E9:74_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#G1" target="_self">GroupMorphismStr</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font> #)
 
<b>by </b><i><a class="txt" href="grcat_1.html#E4:74_1_1_2"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:74_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="grcat_1.html#E5:74_1_1_2"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="grcat_1.html#M4" target="_self">GroupMorphism_DOMAIN</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="grcat_1.html#E7:74_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="grcat_1.html#T38" target="_self">Th38</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E10:74_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> iff <font color="Olive">b<sub>1</sub></font> is  <a href="grcat_1.html#V2" target="_self">strict</a>  <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="grcat_1.html#E6:74_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="grcat_1.html#E8:74_1_1"><i><font color="Green">E41</font></i></a></i>;<br/>


</div>
