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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="grcat_1.html#NM3" title="GRCAT_1:NM.3">Group_DOMAIN</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="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>  st  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Olive" title="b2">f</font> holds <br/><font color="Olive" title="b1">g</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font></span><br/>

<b>set </b><font color="Maroon" title="c2">X</font> =  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>,   <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>] <b>means </b> <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> $2;<br/>
<b>let </b><font color="Maroon" title="c3">g</font>, <font color="Maroon" title="c4">f</font> be    <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">cod</a> <font color="Maroon" title="c4">f</font> implies <font color="Maroon" title="c3">g</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font> )</span><br/>

<b>assume </b><a NAME="E1:86"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">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="c3">g</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">G1</font>, <font color="Maroon" title="c6">G2</font>, <font color="Maroon" title="c7">G3</font> being  <a href="algstr_0.html#V8" title="ALGSTR_0:attr.8">strict</a>  <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:86"/><i><font color="Green" title="E48">A1</font></i>: 
( <font color="Maroon" title="c3">g</font> is    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c6">G2</font>,<font color="Maroon" title="c7">G3</font> &amp; <font color="Maroon" title="c4">f</font> is    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font> )
 <b>by </b><i><a class="ref" href="grcat_1.html#T44" target="_self" title="GRCAT_1:th.44">Th44</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">g'</font> = <font color="Maroon" title="c3">g</font> as    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c6">G2</font>,<font color="Maroon" title="c7">G3</font> <b>by </b><i><a class="txt" href="#E3:86"><i><font color="Green" title="E48">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">f'</font> = <font color="Maroon" title="c4">f</font> as    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font> <b>by </b><i><a class="txt" href="#E3:86"><i><font color="Green" title="E48">A1</font></i></a></i>;<br/>
<a NAME="E6:86"/>
<font color="Maroon" title="c8">g'</font> <a href="grcat_1.html#K15" title="GRCAT_1:func.15">*</a> <font color="Maroon" title="c9">f'</font> is    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c7">G3</font>
 
;<br/>
<b>hence </b><a NAME="E7:86"/>
<font color="Maroon" title="c3">g</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">Morphs</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="grcat_1.html#D30" target="_self" title="GRCAT_1:def.30">Def30</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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