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<a NAME="E1:21_1_1"/>
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds  <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</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">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">w</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="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">w</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</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="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">w</font></span>)</span></span><br/>

<a NAME="E1:21_1_1_1"/>
( <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">v</font>) &amp; <span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c4">w</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D6" target="_self" onmouseover="rs('algstr_4/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E2:21_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_4.html#K3" title="ALGSTR_4:func.3">ClassOp</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">w</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<font color="Maroon" title="c2">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D4" target="_self" onmouseover="rs('algstr_4/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21_1_1_1"/>
<span class="p1">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</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="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D6" target="_self" onmouseover="rs('algstr_4/D6')" onmouseout="rh()">Def6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:21_1_1"/>
 <a href="algstr_4.html#K5" title="ALGSTR_4:func.5">nat_hom</a> <font color="Maroon" title="c2">R</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" onmouseover="rs('group_6/D6')" onmouseout="rh()">GROUP_6:def 6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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