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

<a NAME="E1:13_1_1"/>
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">f0</font> =  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">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="c2">G</font>,<font color="Maroon" title="c3">H</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D26" title="GRCAT_1:def.26">GRCAT_1:def 26</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">D</font> = <span class="p1">{<span class="default"> <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">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="c2">G</font>,<font color="Maroon" title="c3">H</font> : ( <font color="Olive" title="b1">f</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="mod_2.html#V1" title="MOD_2:attr.1">homogeneous</a>  ) </span>}</span> ;<br/>
<a NAME="E3:13_1_1"/>
( <font color="Maroon" title="c4">f0</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  &amp; <font color="Maroon" title="c4">f0</font> is  <a href="mod_2.html#V1" title="MOD_2:attr.1">homogeneous</a>  )
 
;<br/>
<a NAME="E4:13_1_1"/><span class="kw">then </span>
 <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">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="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">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="c2">G</font>,<font color="Maroon" title="c3">H</font> : ( <font color="Olive" title="b1">f</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="mod_2.html#V1" title="MOD_2:attr.1">homogeneous</a>  ) </span>}</span> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">D</font> = <span class="p1">{<span class="default"> <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">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="c2">G</font>,<font color="Maroon" title="c3">H</font> : ( <font color="Olive" title="b1">f</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="mod_2.html#V1" title="MOD_2:attr.1">homogeneous</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="E6:13_1_1"/><span class="lab"><font color="Green" title="E6">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> is   <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">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="c6">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">x</font> is   <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:13_1_1_1"/>
<font color="Maroon" title="c7">x</font> is   <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as   <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> ;<br/>
<a NAME="E3:13_1_1_1"/>
 <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_2.html#D11" title="MOD_2:def.11">MOD_2:def 11</a></span>;<br/>
<a NAME="E4:13_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A2</font></span>: 
the <a href="mod_2.html#U1" title="MOD_2:sel.1">Dom</a> of <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_2.html#D6" title="MOD_2:def.6">MOD_2:def 6</a></span>;<br/>
<a NAME="E5:13_1_1_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
 <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_2.html#D11" title="MOD_2:def.11">MOD_2:def 11</a></span>;<br/>
<a NAME="E6:13_1_1_1"/><span class="kw">then </span>
the <a href="mod_2.html#U2" title="MOD_2:sel.2">Cod</a> of <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_2.html#D7" title="MOD_2:def.7">MOD_2:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = the <a href="mod_2.html#U3" title="MOD_2:sel.3">Fun</a> of <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13_1_1_1"><span class="lab"><font color="Green" title="E7">A2</font></span></a></span>;<br/>
<a NAME="E8:13_1_1_1"/>
<font color="Maroon" title="c9">f</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c9">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="c2">G</font>,<font color="Maroon" title="c3">H</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#D26" title="GRCAT_1:def.26">GRCAT_1:def 26</a></span>;<br/>
<a NAME="E10:13_1_1_1"/>
( the <a href="mod_2.html#U3" title="MOD_2:sel.3">Fun</a> of <font color="Maroon" title="c8">x</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  &amp; the <a href="mod_2.html#U3" title="MOD_2:sel.3">Fun</a> of <font color="Maroon" title="c8">x</font> is  <a href="mod_2.html#V1" title="MOD_2:attr.1">homogeneous</a>  &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c10">g</font> #) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_1_1_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:13_1_1_1"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="mod_2.html#T10" title="MOD_2:th.10">MOD_2:10</a>, <a class="ref" href="mod_2.html#D7" title="MOD_2:def.7">MOD_2:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:13_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:13_1_1"/><span class="lab"><font color="Green" title="E7">A4</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="c6">D</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font> implies <font color="Maroon" title="c7">x</font> is   <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:13_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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"> <font color="Maroon" title="c7">x</font> is   <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font></span><br/>

<a NAME="E2:13_1_1_2"/><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="c2">G</font>,<font color="Maroon" title="c3">H</font> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">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>  &amp; <font color="Olive" title="b1">f</font> is  <a href="mod_2.html#V1" title="MOD_2:attr.1">homogeneous</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:13_1_1_2"/>
<font color="Maroon" title="c7">x</font> is   <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_2.html#T12" title="MOD_2:th.12">MOD_2:12</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="E8:13_1_1"/><span class="kw">then </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="c6">D</font> holds  <font color="Olive" title="b1">x</font> is   <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">D</font> = <font color="Maroon" title="c6">D</font> as    <a href="modcat_1.html#M5" title="MODCAT_1:mode.5">LModMorphism_DOMAIN</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="modcat_1.html#T4" target="_self" title="MODCAT_1:th.4">Th4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">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="c7">D</font> iff <font color="Olive" title="b1">x</font> is   <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> )</span><br/>

<span class="kw">thus </span><a NAME="E10:13_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="c7">D</font> iff <font color="Olive" title="b1">x</font> is   <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:13_1_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E6:13_1_1"><span class="lab"><font color="Green" title="E6">A1</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>
