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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V2">unital</a>  <a href="monoid_0.html#M2" target="_self">SubStr</a> of  <a href="monoid_0.html#K3" target="_self">&lt;REAL,+&gt;</a> ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:105"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="monoid_0.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E3:105"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="monoid_0.html#T25" target="_self">Th25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K9">+</a> 0 = 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E2:105"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="monoid_0.html#T43" target="_self">Th43</a></i>
;<br/>
<a NAME="E6:105"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</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">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:105_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 ;<br/><b>hence </b><a NAME="E2:105_2"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:105"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:105"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 
<b>by </b><i><a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>
<b>hence </b><a NAME="E9:105"/>
 <a href="binop_1.html#K3">the_unity_wrt</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="monoid_0.html#E6:105"><i><font color="Green">E52</font></i></a>, <a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/>


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