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<span class="kw">let </span><font color="Maroon" title="c1">N</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>   <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of  <a href="monoid_0.html#K2" title="MONOID_0:func.2">&lt;REAL,+&gt;</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:101"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
 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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T6" target="_self" title="MONOID_0:th.6">Th6</a></span>;<br/>
<a NAME="E3:101"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T23" target="_self" title="MONOID_0:th.23">Th25</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">N</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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#K4" title="BINOP_1:func.4">.</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> )</span><br/><a NAME="E1:101_1"/>
( <font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#K4" title="BINOP_1:func.4">.</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font>) )
 ;<br/><span class="kw">hence </span><a NAME="E2:101_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">N</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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#K4" title="BINOP_1:func.4">.</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>; <a class="txt" 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>
<a NAME="E6:101"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T3" title="BINOP_1:th.3">BINOP_1:3</a></span>;<br/>
<font color="Maroon" title="c3">x</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> = 
<font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T39" target="_self" title="MONOID_0:th.39">Th43</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:101"/>
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:101"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></span>; <a class="txt" 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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