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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">G2</font>, <font color="Olive" title="b2">G3</font>, <font color="Olive" title="b3">G1</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b4">G</font> being   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Olive" title="b1">G2</font>,<font color="Olive" title="b2">G3</font><br/> for <font color="Olive" title="b5">F</font> being   <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="b1">G2</font><br/> for <font color="Olive" title="b6">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">G2</font>,<font color="Olive" title="b2">G3</font><br/> for <font color="Olive" title="b7">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b3">G1</font>,<font color="Olive" title="b1">G2</font>  st <font color="Olive" title="b4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Olive" title="b1">G2</font>,<font color="Olive" title="b2">G3</font>,<font color="Olive" title="b6">g</font> #) &amp; <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Olive" title="b3">G1</font>,<font color="Olive" title="b1">G2</font>,<font color="Olive" title="b7">f</font> #) holds <br/>( <font color="Olive" title="b4">G</font> <a href="mod_2.html#K9" title="MOD_2:func.9">*'</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Olive" title="b3">G1</font>,<font color="Olive" title="b2">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b6">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b7">f</font></span>)</span> #) &amp; <font color="Olive" title="b4">G</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Olive" title="b3">G1</font>,<font color="Olive" title="b2">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b6">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b7">f</font></span>)</span> #) )</span><br/><b>let </b><font color="Maroon" title="c2">G2</font>, <font color="Maroon" title="c3">G3</font>, <font color="Maroon" title="c4">G1</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</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> being   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font><br/> for <font color="Olive" title="b2">F</font> being   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">G2</font><br/> for <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font><br/> for <font color="Olive" title="b4">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font>,<font color="Olive" title="b3">g</font> #) &amp; <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Olive" title="b4">f</font> #) holds <br/>( <font color="Olive" title="b1">G</font> <a href="mod_2.html#K9" title="MOD_2:func.9">*'</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">f</font></span>)</span> #) &amp; <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#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">f</font></span>)</span> #) )</span><br/><b>let </b><font color="Maroon" title="c5">G</font> be    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</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">F</font> being   <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">G2</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font>,<font color="Olive" title="b2">g</font> #) &amp; <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Olive" title="b3">f</font> #) holds <br/>( <font color="Maroon" title="c5">G</font> <a href="mod_2.html#K9" title="MOD_2:func.9">*'</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">f</font></span>)</span> #) &amp; <font color="Maroon" title="c5">G</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">f</font></span>)</span> #) )</span><br/><b>let </b><font color="Maroon" title="c6">F</font> be    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">G2</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> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font>,<font color="Olive" title="b1">g</font> #) &amp; <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Olive" title="b2">f</font> #) holds <br/>( <font color="Maroon" title="c5">G</font> <a href="mod_2.html#K9" title="MOD_2:func.9">*'</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">f</font></span>)</span> #) &amp; <font color="Maroon" title="c5">G</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">f</font></span>)</span> #) )</span><br/><b>let </b><font color="Maroon" title="c7">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</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">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font>,<font color="Maroon" title="c7">g</font> #) &amp; <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Olive" title="b1">f</font> #) holds <br/>( <font color="Maroon" title="c5">G</font> <a href="mod_2.html#K9" title="MOD_2:func.9">*'</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f</font></span>)</span> #) &amp; <font color="Maroon" title="c5">G</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f</font></span>)</span> #) )</span><br/><b>let </b><font color="Maroon" title="c8">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font>,<font color="Maroon" title="c7">g</font> #) &amp; <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c8">f</font> #) implies ( <font color="Maroon" title="c5">G</font> <a href="mod_2.html#K9" title="MOD_2:func.9">*'</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> #) &amp; <font color="Maroon" title="c5">G</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> #) ) )</span><br/>















<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font>,<font color="Maroon" title="c7">g</font> #) &amp; <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c8">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="c5">G</font> <a href="mod_2.html#K9" title="MOD_2:func.9">*'</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> #) &amp; <font color="Maroon" title="c5">G</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> #) )</span><br/>

 <a href="mod_2.html#K2" title="MOD_2:func.2">dom</a> <font color="Maroon" title="c5">G</font> = 
<font color="Maroon" title="c2">G2</font>
<b>by </b><i><a class="ref" href="mod_2.html#D11" target="_self" title="MOD_2:def.11">Def11</a></i>
<br/>.= 
 <a href="mod_2.html#K3" title="MOD_2:func.3">cod</a> <font color="Maroon" title="c6">F</font>
<b>by </b><i><a class="ref" href="mod_2.html#D11" target="_self" title="MOD_2:def.11">Def11</a></i>
;<br/>
<b>hence </b><a NAME="E3:31"/>
( <font color="Maroon" title="c5">G</font> <a href="mod_2.html#K9" title="MOD_2:func.9">*'</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> #) &amp; <font color="Maroon" title="c5">G</font> <a href="mod_2.html#K8" title="MOD_2:func.8">*</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c3">G3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> #) )
 <b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="mod_2.html#D13" target="_self" title="MOD_2:def.13">Def13</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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