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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">defpred </span><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>] <span class="kw">means </span> <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/>
<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:33"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:33"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_2.html#T16" target="_self" title="MOD_2:th.16">Th16</a></span>;<br/>
<a NAME="E4:33"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="mod_2.html#D11" target="_self" title="MOD_2:def.11">Def11</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:33"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_2.html#T16" target="_self" title="MOD_2:th.16">Th16</a></span>;<br/>
<a NAME="E7:33"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="mod_2.html#D11" target="_self" title="MOD_2:def.11">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:33"/>
 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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E16">A4</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/>


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