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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">g</font>, <font color="Olive" title="b2">f</font> being  <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="b1">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="b2">f</font> holds <br/> ex <font color="Olive" title="b3">G1</font>, <font color="Olive" title="b4">G2</font>, <font color="Olive" title="b5">G3</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">g</font> is    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Olive" title="b4">G2</font>,<font color="Olive" title="b5">G3</font> &amp; <font color="Olive" title="b2">f</font> is    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Olive" title="b3">G1</font>,<font color="Olive" title="b4">G2</font> )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="mod_2.html#NM1" title="MOD_2:NM.1">LModMorphism</a> of <font color="Maroon" title="c1">R</font>,  <a href="mod_2.html#NM1" title="MOD_2:NM.1">LModMorphism</a> of <font color="Maroon" title="c1">R</font>] <b>means </b> <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> $2;<br/>
<b>let </b><font color="Maroon" title="c2">g</font>, <font color="Maroon" title="c3">f</font> be   <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="c2">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="c3">f</font> implies  ex <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font>, <font color="Olive" title="b3">G3</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Maroon" title="c2">g</font> is    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font> &amp; <font color="Maroon" title="c3">f</font> is    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> ) )</span><br/>

<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">f</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font>, <font color="Olive" title="b3">G3</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Maroon" title="c2">g</font> is    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font> &amp; <font color="Maroon" title="c3">f</font> is    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c4">G2</font>, <font color="Maroon" title="c5">G3</font> being   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c2">g</font> is    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c4">G2</font>,<font color="Maroon" title="c5">G3</font>
 <b>by </b><i><a class="ref" href="mod_2.html#T16" target="_self" title="MOD_2:th.16">Th16</a></i>;<br/>
<a NAME="E4:27"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="txt" href="mod_2.html#E3:27"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="mod_2.html#D11" target="_self" title="MOD_2:def.11">Def11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">G1</font>, <font color="Maroon" title="c7">G2</font> being   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E6:27"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c3">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="mod_2.html#T16" target="_self" title="MOD_2:th.16">Th16</a></i>;<br/>
<a NAME="E7:27"/>
<font color="Maroon" title="c7">G2</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">f</font>
 
<b>by </b><i><a class="txt" href="mod_2.html#E6:27"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="mod_2.html#D11" target="_self" title="MOD_2:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E8:27"/>
 ex <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font>, <font color="Olive" title="b3">G3</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Maroon" title="c2">g</font> is    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font> &amp; <font color="Maroon" title="c3">f</font> is    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> )
 <b>by </b><i><a class="txt" href="mod_2.html#E1:27"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="mod_2.html#E3:27"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="mod_2.html#E4:27"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="mod_2.html#E6:27"><i><font color="Green" title="E19">A4</font></i></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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