<?xml version="1.0"?>
<div class="add">

<span class="kw">consider </span><font color="Maroon" title="c4">G19</font>, <font color="Maroon" title="c5">G29</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="E2:34_1_2"/><span class="lab"><font color="Green" title="E14">A8</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="c4">G19</font>,<font color="Maroon" title="c5">G29</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">F9</font> = <font color="Maroon" title="c3">F</font> as    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c4">G19</font>,<font color="Maroon" title="c5">G29</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">G2</font>, <font color="Maroon" title="c8">G39</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="E5:34_1_2"/><span class="lab"><font color="Green" title="E15">A9</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="c7">G2</font>,<font color="Maroon" title="c8">G39</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="E6:34_1_2"/>
<font color="Maroon" title="c7">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="#E5:34_1_2"><span class="lab"><font color="Green" title="E15">A9</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">F9</font> = <font color="Maroon" title="c6">F9</font> as    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c4">G19</font>,<font color="Maroon" title="c7">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E13">A1</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="c10">f9</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">G19</font>,<font color="Maroon" title="c7">G2</font><span class="kw"> such that </span><br/><a NAME="E9:34_1_2"/><span class="lab"><font color="Green" title="E16">A10</font></span>: 
 <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# the <a href="mod_2.html#U1" title="MOD_2:sel.1">Dom</a> of <font color="Maroon" title="c9">F9</font>,the <a href="mod_2.html#U2" title="MOD_2:sel.2">Cod</a> of <font color="Maroon" title="c9">F9</font>,the <a href="mod_2.html#U3" title="MOD_2:sel.3">Fun</a> of <font color="Maroon" title="c9">F9</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">G19</font>,<font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">f9</font> #)
 <span class="kw">and </span><br/><a NAME="E10:34_1_2"/>
( <font color="Maroon" title="c10">f9</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  &amp; <font color="Maroon" title="c10">f9</font> is  <a href="mod_2.html#V1" title="MOD_2:attr.1">homogeneous</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_2.html#T14" target="_self" title="MOD_2:th.14">Th14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">G9</font> = <font color="Maroon" title="c2">G</font> as    <a href="mod_2.html#M1" title="MOD_2:mode.1">Morphism</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c8">G39</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_1_2"><span class="lab"><font color="Green" title="E15">A9</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">S1</font>, <font color="Maroon" title="c13">S2</font> be   <a href="mod_2.html#V2" title="MOD_2:attr.2">strict</a>  <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"> ( ( for <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><br/> for <font color="Olive" title="b4">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>  st  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# the <a href="mod_2.html#U1" title="MOD_2:sel.1">Dom</a> of <font color="Maroon" title="c2">G</font>,the <a href="mod_2.html#U2" title="MOD_2:sel.2">Cod</a> of <font color="Maroon" title="c2">G</font>,the <a href="mod_2.html#U3" title="MOD_2:sel.3">Fun</a> of <font color="Maroon" title="c2">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="b2">G2</font>,<font color="Olive" title="b3">G3</font>,<font color="Olive" title="b4">g</font> #) &amp;  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# the <a href="mod_2.html#U1" title="MOD_2:sel.1">Dom</a> of <font color="Maroon" title="c3">F</font>,the <a href="mod_2.html#U2" title="MOD_2:sel.2">Cod</a> of <font color="Maroon" title="c3">F</font>,the <a href="mod_2.html#U3" title="MOD_2:sel.3">Fun</a> of <font color="Maroon" title="c3">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="b1">G1</font>,<font color="Olive" title="b2">G2</font>,<font color="Olive" title="b5">f</font> #) holds <br/><font color="Maroon" title="c12">S1</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">G1</font>,<font color="Olive" title="b3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">f</font></span>)</span> #) ) &amp; ( for <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><br/> for <font color="Olive" title="b4">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>  st  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# the <a href="mod_2.html#U1" title="MOD_2:sel.1">Dom</a> of <font color="Maroon" title="c2">G</font>,the <a href="mod_2.html#U2" title="MOD_2:sel.2">Cod</a> of <font color="Maroon" title="c2">G</font>,the <a href="mod_2.html#U3" title="MOD_2:sel.3">Fun</a> of <font color="Maroon" title="c2">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="b2">G2</font>,<font color="Olive" title="b3">G3</font>,<font color="Olive" title="b4">g</font> #) &amp;  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# the <a href="mod_2.html#U1" title="MOD_2:sel.1">Dom</a> of <font color="Maroon" title="c3">F</font>,the <a href="mod_2.html#U2" title="MOD_2:sel.2">Cod</a> of <font color="Maroon" title="c3">F</font>,the <a href="mod_2.html#U3" title="MOD_2:sel.3">Fun</a> of <font color="Maroon" title="c3">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="b1">G1</font>,<font color="Olive" title="b2">G2</font>,<font color="Olive" title="b5">f</font> #) holds <br/><font color="Maroon" title="c13">S2</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">G1</font>,<font color="Olive" title="b3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">f</font></span>)</span> #) ) implies <font color="Maroon" title="c12">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">S2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E12:34_1_2"/><span class="lab"><font color="Green" title="E17">A11</font></span>: 
for <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><br/> for <font color="Olive" title="b4">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>  st  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# the <a href="mod_2.html#U1" title="MOD_2:sel.1">Dom</a> of <font color="Maroon" title="c2">G</font>,the <a href="mod_2.html#U2" title="MOD_2:sel.2">Cod</a> of <font color="Maroon" title="c2">G</font>,the <a href="mod_2.html#U3" title="MOD_2:sel.3">Fun</a> of <font color="Maroon" title="c2">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="b2">G2</font>,<font color="Olive" title="b3">G3</font>,<font color="Olive" title="b4">g</font> #) &amp;  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# the <a href="mod_2.html#U1" title="MOD_2:sel.1">Dom</a> of <font color="Maroon" title="c3">F</font>,the <a href="mod_2.html#U2" title="MOD_2:sel.2">Cod</a> of <font color="Maroon" title="c3">F</font>,the <a href="mod_2.html#U3" title="MOD_2:sel.3">Fun</a> of <font color="Maroon" title="c3">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="b1">G1</font>,<font color="Olive" title="b2">G2</font>,<font color="Olive" title="b5">f</font> #) holds <br/><font color="Maroon" title="c12">S1</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">G1</font>,<font color="Olive" title="b3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">f</font></span>)</span> #)
 <span class="kw">and </span><br/><a NAME="E13:34_1_2"/><span class="lab"><font color="Green" title="E18">A12</font></span>: 
for <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><br/> for <font color="Olive" title="b4">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">G2</font>,<font color="Olive" title="b3">G3</font><br/> for <font color="Olive" title="b5">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>  st  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# the <a href="mod_2.html#U1" title="MOD_2:sel.1">Dom</a> of <font color="Maroon" title="c2">G</font>,the <a href="mod_2.html#U2" title="MOD_2:sel.2">Cod</a> of <font color="Maroon" title="c2">G</font>,the <a href="mod_2.html#U3" title="MOD_2:sel.3">Fun</a> of <font color="Maroon" title="c2">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="b2">G2</font>,<font color="Olive" title="b3">G3</font>,<font color="Olive" title="b4">g</font> #) &amp;  <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# the <a href="mod_2.html#U1" title="MOD_2:sel.1">Dom</a> of <font color="Maroon" title="c3">F</font>,the <a href="mod_2.html#U2" title="MOD_2:sel.2">Cod</a> of <font color="Maroon" title="c3">F</font>,the <a href="mod_2.html#U3" title="MOD_2:sel.3">Fun</a> of <font color="Maroon" title="c3">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="b1">G1</font>,<font color="Olive" title="b2">G2</font>,<font color="Olive" title="b5">f</font> #) holds <br/><font color="Maroon" title="c13">S2</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">G1</font>,<font color="Olive" title="b3">G3</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">f</font></span>)</span> #)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">S2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c14">g9</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c8">G39</font><span class="kw"> such that </span><br/><a NAME="E15:34_1_2"/><span class="lab"><font color="Green" title="E19">A13</font></span>: 
 <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# the <a href="mod_2.html#U1" title="MOD_2:sel.1">Dom</a> of <font color="Maroon" title="c11">G9</font>,the <a href="mod_2.html#U2" title="MOD_2:sel.2">Cod</a> of <font color="Maroon" title="c11">G9</font>,the <a href="mod_2.html#U3" title="MOD_2:sel.3">Fun</a> of <font color="Maroon" title="c11">G9</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="c7">G2</font>,<font color="Maroon" title="c8">G39</font>,<font color="Maroon" title="c14">g9</font> #)
 <span class="kw">and </span><br/><a NAME="E16:34_1_2"/>
( <font color="Maroon" title="c14">g9</font> is  <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>  &amp; <font color="Maroon" title="c14">g9</font> is  <a href="mod_2.html#V1" title="MOD_2:attr.1">homogeneous</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_2.html#T14" target="_self" title="MOD_2:th.14">Th14</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c12">S1</font> = 
 <a href="mod_2.html#G1" title="MOD_2:aggr.1">LModMorphismStr</a>(# <font color="Maroon" title="c4">G19</font>,<font color="Maroon" title="c8">G39</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">g9</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">f9</font></span>)</span> #)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:34_1_2"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="txt" href="#E15:34_1_2"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="txt" href="#E9:34_1_2"><span class="lab"><font color="Green" title="E16">A10</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">S2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:34_1_2"><span class="lab"><font color="Green" title="E18">A12</font></span></a>, <a class="txt" href="#E15:34_1_2"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="txt" href="#E9:34_1_2"><span class="lab"><font color="Green" title="E16">A10</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/>


</div>
