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

<span class="kw">reconsider </span><font color="Maroon" title="c3">f0</font> =  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">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#K16" title="GRCAT_1:func.16">Maps</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">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#K16" title="GRCAT_1:func.16">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="E2:81_1_1"/>
<font color="Maroon" title="c3">f0</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a> 
 
;<br/>
<a NAME="E3:81_1_1"/>
 <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="c3">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#K16" title="GRCAT_1:func.16">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/>
<span class="kw">then </span><span class="kw">reconsider </span><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#K16" title="GRCAT_1:func.16">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="E5:81_1_1"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
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="81_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:81_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E2:81_1_1_1"/><span class="kw">then </span>
 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#K16" title="GRCAT_1:func.16">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/>
<span class="kw">hence </span><a NAME="E3:81_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#T20" target="_self" title="GRCAT_1:th.20">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:81_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A2</font></span>: 
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="E7:81_1_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
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="81_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:81_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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="E3:81_1_1_2"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
(  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">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#K9" title="GRCAT_1:func.9">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D19" target="_self" title="GRCAT_1:def.19">Def19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> ;<br/>
<span class="kw">reconsider </span><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#K16" title="GRCAT_1:func.16">Maps</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<a NAME="E6:81_1_1_2"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<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> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81_1_1_2"><span class="lab"><font color="Green" title="E33">A4</font></span></a></span>;<br/>
<a NAME="E7:81_1_1_2"/>
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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#T18" target="_self" title="GRCAT_1:th.18">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:81_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81_1_1_2"><span class="lab"><font color="Green" title="E34">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="grcat_1.html#T38" target="_self" title="GRCAT_1:th.38">Th38</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">D</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">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> )</span><br/>

<span class="kw">thus </span><a NAME="E9:81_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:81_1_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E7:81_1_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
