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<b>let </b><font color="Maroon" title="c1">K</font>, <font color="Maroon" title="c2">L</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">J</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b2">V</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b3">W</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c2">L</font> holds   <a href="mod_4.html#K8" title="MOD_4:func.8">ZeroMap</a> <font color="Olive" title="b2">V</font>,<font color="Olive" title="b3">W</font> is    <a href="mod_4.html#M2" title="MOD_4:mode.2">Homomorphism</a> of <font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">V</font>,<font color="Olive" title="b3">W</font></span><br/><b>let </b><font color="Maroon" title="c3">J</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">L</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">V</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">W</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c2">L</font> holds   <a href="mod_4.html#K8" title="MOD_4:func.8">ZeroMap</a> <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font> is    <a href="mod_4.html#M2" title="MOD_4:mode.2">Homomorphism</a> of <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font></span><br/><b>let </b><font color="Maroon" title="c4">V</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</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">W</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c2">L</font> holds   <a href="mod_4.html#K8" title="MOD_4:func.8">ZeroMap</a> <font color="Maroon" title="c4">V</font>,<font color="Olive" title="b1">W</font> is    <a href="mod_4.html#M2" title="MOD_4:mode.2">Homomorphism</a> of <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c4">V</font>,<font color="Olive" title="b1">W</font></span><br/><b>let </b><font color="Maroon" title="c5">W</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="mod_4.html#K8" title="MOD_4:func.8">ZeroMap</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c5">W</font> is    <a href="mod_4.html#M2" title="MOD_4:mode.2">Homomorphism</a> of <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c5">W</font></span><br/>









<b>set </b><font color="Maroon" title="c6">z</font> =  <a href="mod_4.html#K8" title="MOD_4:func.8">ZeroMap</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c5">W</font>;<br/>
<a NAME="E1:92"/><i><font color="Green" title="E55">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">V</font> holds  <span class="p1">(<span class="default"><a href="mod_4.html#K8" title="MOD_4:func.8">ZeroMap</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c5">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K8" title="MOD_4:func.8">ZeroMap</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c5">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K8" title="MOD_4:func.8">ZeroMap</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c5">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mod_4.html#E86"><i><font color="Green" title="E52">Lm18</font></i></a></i>;<br/>
<a NAME="E2:92"/>
for <font color="Olive" title="b1">a</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">V</font> holds  <span class="p1">(<span class="default"><a href="mod_4.html#K8" title="MOD_4:func.8">ZeroMap</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c5">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mod_4.html#K8" title="MOD_4:func.8">ZeroMap</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c5">W</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mod_4.html#E87"><i><font color="Green" title="E53">Lm19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:92"/>
 <a href="mod_4.html#K8" title="MOD_4:func.8">ZeroMap</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c5">W</font> is    <a href="mod_4.html#M2" title="MOD_4:mode.2">Homomorphism</a> of <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c5">W</font>
 <b>by </b><i><a class="txt" href="#E1:92"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="mod_4.html#D24" target="_self" title="MOD_4:def.24">Def24</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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