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<b>let </b><font color="Maroon" title="c1">N</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of  <a href="monoid_0.html#K3" title="MONOID_0:func.3">&lt;REAL,+&gt;</a> ;<br/>

<b>consider </b><font color="Maroon" title="c2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E2:105"/><i><font color="Green" title="E51">A1</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> holds <br/> ( <font color="Maroon" title="c2">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> )
 <b>by </b><i><a class="ref" href="monoid_0.html#T6" target="_self" title="MONOID_0:th.6">Th6</a></i>;<br/>
<a NAME="E3:105"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">N</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="monoid_0.html#T25" target="_self" title="MONOID_0:th.25">Th25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<font color="Maroon" title="c3">x</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 0 = 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E2:105"><i><font color="Green" title="E51">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="ref" href="monoid_0.html#T43" target="_self" title="MONOID_0:th.43">Th43</a></i>
;<br/>
<a NAME="E6:105"/><b>then </b><i><font color="Green" title="E52">A2</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>;<br/><a NAME="E1:105_2"/>
( <font color="Maroon" title="c2">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">N</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">N</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font> )
 ;<br/><b>hence </b><a NAME="E2:105_2"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">N</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">N</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:105"><i><font color="Green" title="E51">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:105"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">N</font>)
 
<b>by </b><i><a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>;<br/>
<b>hence </b><a NAME="E9:105"/>
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="monoid_0.html#E6:105"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></i>;<br/>


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