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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">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/> ex <font color="Olive" title="b4">G1</font>, <font color="Olive" title="b5">G2</font>, <font color="Olive" title="b6">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="Olive" title="b1">V</font> st <br/>( <font color="Olive" title="b2">g</font> is    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Olive" title="b5">G2</font>,<font color="Olive" title="b6">G3</font> &amp; <font color="Olive" title="b3">f</font> is    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Olive" title="b4">G1</font>,<font color="Olive" title="b5">G2</font> )</span><br/><span class="kw">let </span><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:()"><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="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/> 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_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> 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">set </span><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/>
<span class="kw">defpred </span><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>] <span class="kw">means </span> <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/>
<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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  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_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> st <br/>( <font color="Maroon" title="c4">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="c5">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="E1:27"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<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:()"><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_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> st <br/>( <font color="Maroon" title="c4">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="c5">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="c6">G2</font>, <font color="Maroon" title="c7">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><span class="kw"> such that </span><br/><a NAME="E3:27"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c4">g</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="c6">G2</font>,<font color="Maroon" title="c7">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modcat_1.html#D7" target="_self" title="MODCAT_1:def.7">Def7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">G1</font>, <font color="Maroon" title="c9">G29</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><span class="kw"> such that </span><br/><a NAME="E5:27"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c5">f</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="c8">G1</font>,<font color="Maroon" title="c9">G29</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modcat_1.html#D7" target="_self" title="MODCAT_1:def.7">Def7</a></span>;<br/>
<a NAME="E6:27"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c9">G29</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="mod_2.html#D11" title="MOD_2:def.11">MOD_2:def 11</a></span>;<br/>
<a NAME="E7:27"/>
<font color="Maroon" title="c6">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modcat_1.html#K5" title="MODCAT_1:func.5">dom'</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="mod_2.html#D11" title="MOD_2:def.11">MOD_2:def 11</a></span>;<br/>
<span class="kw">hence </span><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_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> st <br/>( <font color="Maroon" title="c4">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="c5">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="#E1:27"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E21">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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