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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a> <a href="mod_2.html#NM1" title="MOD_2:NM.1">LModMorphism</a> of <font color="Maroon" title="c1">R</font>  st  <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Olive" title="b2">g</font> &amp;  <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b3">h</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">h</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font>, <font color="Maroon" title="c4">h</font> be  <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a> <a href="mod_2.html#NM1" title="MOD_2:NM.1">LModMorphism</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c3">g</font> &amp;  <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c2">f</font> implies <font color="Maroon" title="c4">h</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c2">f</font> )</span><br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E21">A1</font></i>: 
(  <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c3">g</font> &amp;  <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c2">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">h</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c2">f</font></span><br/>

<b>set </b><font color="Maroon" title="c5">G2</font> =  <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c2">f</font>;<br/>
<b>set </b><font color="Maroon" title="c6">G3</font> =  <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c3">g</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">f'</font> = <font color="Maroon" title="c2">f</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  <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Maroon" title="c2">f</font>, <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c2">f</font> <b>by </b><i><a class="ref" href="mod_2.html#D11" target="_self" title="MOD_2:def.11">Def11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">g'</font> = <font color="Maroon" title="c3">g</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  <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c2">f</font>, <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c3">g</font> <b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="mod_2.html#D11" target="_self" title="MOD_2:def.11">Def11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">h'</font> = <font color="Maroon" title="c4">h</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  <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c3">g</font>, <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c4">h</font> <b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="mod_2.html#D11" target="_self" title="MOD_2:def.11">Def11</a></i>;<br/>
<a NAME="E5:35"/>
<font color="Maroon" title="c9">h'</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g'</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c7">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h'</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c8">g'</font></span>)</span> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c7">f'</font>
 
<b>by </b><i><a class="ref" href="mod_2.html#T24" target="_self" title="MOD_2:th.24">Th24</a></i>;<br/>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon" title="c4">h</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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