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

<b>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <a href="mod_2.html#G1">LModMorphismStr</a>(# <font color="Maroon">G</font>,<font color="Maroon">H</font>,<font color="Olive">f</font> #) where <font color="Olive">f</font> is    <a href="grcat_1.html#M6">Element</a> of  <a href="grcat_1.html#K17">Maps</a> <font color="Maroon">G</font>,<font color="Maroon">H</font> : <font color="Olive">f</font> is <a href="mod_2.html#V1">linear</a> </span>}</span> ;<br/>
<a NAME="E1:11_1_1"/>
 <a href="grcat_1.html#K8">ZeroMap</a> <font color="Maroon">G</font>,<font color="Maroon">H</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">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</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">f0</font> =  <a href="grcat_1.html#K8">ZeroMap</a> <font color="Maroon">G</font>,<font color="Maroon">H</font> as    <a href="grcat_1.html#M6">Element</a> of  <a href="grcat_1.html#K17">Maps</a> <font color="Maroon">G</font>,<font color="Maroon">H</font> <b>by </b><i><a class="ref" href="grcat_1.html#D26">GRCAT_1:def 26</a></i>;<br/>
<a NAME="E3:11_1_1"/>
<font color="Maroon">f0</font> is <a href="mod_2.html#V1">linear</a>
 
<b>by </b><i><a class="ref" href="mod_2.html#T8">MOD_2:8</a></i>;<br/>
<a NAME="E4:11_1_1"/><b>then </b>
 <a href="mod_2.html#G1">LModMorphismStr</a>(# <font color="Maroon">G</font>,<font color="Maroon">H</font>,<font color="Maroon">f0</font> #) <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <a href="mod_2.html#G1">LModMorphismStr</a>(# <font color="Maroon">G</font>,<font color="Maroon">H</font>,<font color="Olive">f</font> #) where <font color="Olive">f</font> is    <a href="grcat_1.html#M6">Element</a> of  <a href="grcat_1.html#K17">Maps</a> <font color="Maroon">G</font>,<font color="Maroon">H</font> : <font color="Olive">f</font> is <a href="mod_2.html#V1">linear</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <a href="mod_2.html#G1">LModMorphismStr</a>(# <font color="Maroon">G</font>,<font color="Maroon">H</font>,<font color="Olive">f</font> #) where <font color="Olive">f</font> is    <a href="grcat_1.html#M6">Element</a> of  <a href="grcat_1.html#K17">Maps</a> <font color="Maroon">G</font>,<font color="Maroon">H</font> : <font color="Olive">f</font> is <a href="mod_2.html#V1">linear</a> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E6:11_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/><font color="Olive">x</font> is  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">Morphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:11_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:11_1_1_1"/><b>then </b>
 ex <font color="Olive">f</font> being   <a href="grcat_1.html#M6">Element</a> of  <a href="grcat_1.html#K17">Maps</a> <font color="Maroon">G</font>,<font color="Maroon">H</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="mod_2.html#G1">LModMorphismStr</a>(# <font color="Maroon">G</font>,<font color="Maroon">H</font>,<font color="Olive">f</font> #) &amp; <font color="Olive">f</font> is <a href="mod_2.html#V1">linear</a> )
 
;<br/>
<b>hence </b><a NAME="E3:11_1_1_1"/>
<font color="Maroon">x</font> is  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">Morphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="mod_2.html#T12">MOD_2:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:11_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <font color="Olive">x</font> is  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">Morphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>
 
;<br/>
<a NAME="E8:11_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> is  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">Morphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:11_1_1_2"/>
<font color="Maroon">x</font> is  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">Morphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">Morphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font> ;<br/>
<a NAME="E3:11_1_1_2"/>
(  <a href="mod_2.html#K2">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> &amp;  <a href="mod_2.html#K3">cod</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="mod_2.html#D11">MOD_2:def 11</a></i>;<br/>
<a NAME="E4:11_1_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
( the <a href="mod_2.html#U1">Dom</a> of <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> &amp; the <a href="mod_2.html#U2">Cod</a> of <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="mod_2.html#D6">MOD_2:def 6</a>, <a class="ref" href="mod_2.html#D7">MOD_2:def 7</a></i>;<br/>
<a NAME="E5:11_1_1_2"/><i><font color="Green">E40</font></i>: 
the <a href="mod_2.html#U3">Fun</a> of <font color="Maroon">x</font> is <a href="mod_2.html#V1">linear</a>
 
<b>by </b><i><a class="ref" href="mod_2.html#T10">MOD_2:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = the <a href="mod_2.html#U3">Fun</a> of <font color="Maroon">x</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font> <b>by </b><i/>;<br/>
<a NAME="E7:11_1_1_2"/>
<font color="Maroon">f</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">G</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</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">g</font> = <font color="Maroon">f</font> as    <a href="grcat_1.html#M6">Element</a> of  <a href="grcat_1.html#K17">Maps</a> <font color="Maroon">G</font>,<font color="Maroon">H</font> <b>by </b><i><a class="ref" href="grcat_1.html#D26">GRCAT_1:def 26</a></i>;<br/>
<a NAME="E9:11_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="mod_2.html#G1">LModMorphismStr</a>(# <font color="Maroon">G</font>,<font color="Maroon">H</font>,<font color="Maroon">g</font> #)
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:11_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as    <a href="modcat_1.html#M5" target="_self">LModMorphism_DOMAIN</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font> <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">D</font>
;<br/>

<b>thus </b><a NAME="E10:11_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> iff <font color="Olive">x</font> is  <a href="mod_2.html#V2">strict</a>  <a href="mod_2.html#M1">Morphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font> )
 <b>by </b><i>, </i>;<br/>


</div>
