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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">V</font> being   <a href="modcat_1.html#M1" title="MODCAT_1:mode.1">LeftMod_DOMAIN</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">f</font> being   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Olive" title="b1">V</font>  st  <a href="modcat_1.html#K5" title="MODCAT_1:func.5">dom'</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modcat_1.html#K6" title="MODCAT_1:func.6">cod'</a> <font color="Olive" title="b3">f</font> holds <br/><font color="Olive" title="b2">g</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Olive" title="b1">V</font></span><br/><b>let </b><font color="Maroon" title="c2">V</font> be    <a href="modcat_1.html#M1" title="MODCAT_1:mode.1">LeftMod_DOMAIN</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"> for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">f</font> being   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font>  st  <a href="modcat_1.html#K5" title="MODCAT_1:func.5">dom'</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modcat_1.html#K6" title="MODCAT_1:func.6">cod'</a> <font color="Olive" title="b2">f</font> holds <br/><font color="Olive" title="b1">g</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font></span><br/>



<b>set </b><font color="Maroon" title="c3">X</font> =  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font>,   <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font>] <b>means </b> <a href="modcat_1.html#K5" title="MODCAT_1:func.5">dom'</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modcat_1.html#K6" title="MODCAT_1:func.6">cod'</a> $2;<br/>
<b>let </b><font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c5">f</font> be    <a href="modcat_1.html#M4" title="MODCAT_1:mode.4">Element</a> of  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="modcat_1.html#K5" title="MODCAT_1:func.5">dom'</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modcat_1.html#K6" title="MODCAT_1:func.6">cod'</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c4">g</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font> )</span><br/>

<b>assume </b><a NAME="E1:26"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c5">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">g</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font></span><br/>

<b>then consider </b><font color="Maroon" title="c6">G1</font>, <font color="Maroon" title="c7">G2</font>, <font color="Maroon" title="c8">G3</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="modcat_1.html#M2" title="MODCAT_1:mode.2">Element</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:26"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c4">g</font> is    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c8">G3</font> &amp; <font color="Maroon" title="c5">f</font> is    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font> )
 <b>by </b><i><a class="ref" href="modcat_1.html#T10" target="_self" title="MODCAT_1:th.10">Th10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">g'</font> = <font color="Maroon" title="c4">g</font> as    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c8">G3</font> <b>by </b><i><a class="txt" href="modcat_1.html#E3:26"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f'</font> = <font color="Maroon" title="c5">f</font> as    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font> <b>by </b><i><a class="txt" href="modcat_1.html#E3:26"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<a NAME="E6:26"/>
<font color="Maroon" title="c9">g'</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c10">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g'</font> <a href="mod_2.html#K9" title="MOD_2:func.9">*'</a> <font color="Maroon" title="c10">f'</font>
 
<b>by </b><i><a class="ref" href="mod_2.html#D14" title="MOD_2:def.14">MOD_2:def 14</a></i>;<br/>
<b>hence </b><a NAME="E7:26"/>
<font color="Maroon" title="c4">g</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modcat_1.html#K4" title="MODCAT_1:func.4">Morphs</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="modcat_1.html#D7" target="_self" title="MODCAT_1:def.7">Def7</a></i>; <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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