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<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">G</font><b> such that </b><br/><a NAME="E2:59_1_1"/><i><font color="Green" title="E31">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">G</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> )
 <b>by </b><i><a class="ref" href="monoid_0.html#T6" target="_self" title="MONOID_0:th.6">Th6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c4">H</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c3">f</font> #);<br/>
<i><font color="Green" title="E32">X</font></i>: <font color="Maroon" title="c3">f</font> = 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">.--&gt;</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="monoid_0.html#T3" target="_self" title="MONOID_0:th.3">Th3</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">a</font>

;<br/>
<a NAME="E5:59_1_1"/><i><font color="Green" title="E33">Y</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="txt" href="monoid_0.html#E2:59_1_1"><i><font color="Green" title="E31">A1</font></i></a></i>;<br/>
<a NAME="E6:59_1_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a</font>,<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">G</font>) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:59_1_1"/><b>then </b>
<font color="Maroon">H<sub>2</sub></font>( <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c3">f</font> #)) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>)
 
<b>by </b><i><a class="ref" href="funct_4.html#T8" title="FUNCT_4:th.8">FUNCT_4:8</a>, <a class="txt" href="monoid_0.html#E4:59_1_1"><i><font color="Green" title="E32">X</font></i></a>, <a class="txt" href="monoid_0.html#E5:59_1_1"><i><font color="Green" title="E33">Y</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">H</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c3">f</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="monoid_0.html#D23" target="_self" title="MONOID_0:def.23">Def23</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">H</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">H</font> is <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a> &amp; <font color="Maroon" title="c5">H</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Maroon" title="c5">H</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp; <font color="Maroon" title="c5">H</font> is <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a> &amp; <font color="Maroon" title="c5">H</font> is <a href="monoid_0.html#V10" title="MONOID_0:attr.10">idempotent</a> &amp; <font color="Maroon" title="c5">H</font> is <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a> &amp; <font color="Maroon" title="c5">H</font> is <a href="monoid_0.html#V17" title="MONOID_0:attr.17">uniquely-decomposable</a> &amp; <font color="Maroon" title="c5">H</font> is <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> )</span><br/>

<b>thus </b><a NAME="E9:59_1_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">H</font>) is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">H</font>) is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">H</font>) is <a href="monoid_0.html#V8" title="MONOID_0:attr.8">cancelable</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">H</font>) is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">H</font>) is <a href="monoid_0.html#V5" title="MONOID_0:attr.5">invertible</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">H</font>) is <a href="monoid_0.html#V9" title="MONOID_0:attr.9">uniquely-decomposable</a> )
 <b>by </b><i><a class="ref" href="monoid_0.html#T3" target="_self" title="MONOID_0:th.3">Th3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D10" target="_self" title="MONOID_0:def.10">MONOID_0:def 10</a>,<a class="ref" href="monoid_0.html#D11" target="_self" title="MONOID_0:def.11">MONOID_0:def 11</a>,<a class="ref" href="monoid_0.html#D12" target="_self" title="MONOID_0:def.12">MONOID_0:def 12</a>,<a class="ref" href="monoid_0.html#D13" target="_self" title="MONOID_0:def.13">MONOID_0:def 13</a>,<a class="ref" href="monoid_0.html#D16" target="_self" title="MONOID_0:def.16">MONOID_0:def 16</a>,<a class="ref" href="monoid_0.html#D19" target="_self" title="MONOID_0:def.19">MONOID_0:def 19</a>,<a class="ref" href="monoid_0.html#D20" target="_self" title="MONOID_0:def.20">MONOID_0:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">H</font> is <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a></span><br/>

<b>thus </b><a NAME="E10:59_1_1"/>
<font color="Maroon" title="c5">H</font> is <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>
 ; <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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