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<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">G</font><span class="kw"> such that </span><br/><a NAME="E2:59_1_1"/><span class="lab"><font color="Green" title="E32">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">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> )
 <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="E3:59_1_1"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">f</font> =  the   <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/>
<span class="kw">set </span><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>, the   <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/>
<a NAME="E4:59_1_1"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
 the   <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> = 
(<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T3" target="_self" title="MONOID_0:th.3">Th3</a></span>
<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="E6:59_1_1"/><span class="kw">then </span>
<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>, the   <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> #)) <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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E4:59_1_1"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="funct_4.html#T7" title="FUNCT_4:th.7">FUNCT_4:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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>, the   <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> #) 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> <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D23" target="_self" title="MONOID_0:def.23">Def23</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">H</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" 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/>

<span class="kw">thus </span><a NAME="E8:59_1_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">H</font>) is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T3" target="_self" title="MONOID_0:th.3">Th3</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E9:59_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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