<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>  <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="monoid_0.html#M3" title="MONOID_0:mode.3">MonoidalSubStr</a> of <font color="Maroon" title="c1">M</font> holds  <font color="Olive" title="b1">N</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a></span><br/><b>let </b><font color="Maroon" title="c2">N</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="monoid_0.html#M3" title="MONOID_0:mode.3">MonoidalSubStr</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">N</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a></span><br/>



<a NAME="E1:79"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c2">N</font> is    <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="monoid_0.html#T23" target="_self" title="MONOID_0:th.23">Th23</a></i>;<br/>
<a NAME="E2:79"/><i><font color="Green" title="E42">A2</font></i>: 
( <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c2">N</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">M</font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">M</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">M</font>) )
 
<b>by </b><i><a class="ref" href="monoid_0.html#D21" target="_self" title="MONOID_0:def.21">Def21</a>, <a class="ref" href="monoid_0.html#D24" target="_self" title="MONOID_0:def.24">Def24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">N</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c2">N</font>),<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">N</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c2">N</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> )</span><br/><a NAME="E1:79_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">N</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">M</font>)
 <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="c4">a'</font> = <font color="Maroon" title="c3">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">N</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c2">N</font>),<font color="Maroon" title="c3">a</font> = 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c2">N</font>) <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">a</font>

<br/>.= 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">M</font>) <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a'</font>
<b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E2:79"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="monoid_0.html#T27" target="_self" title="MONOID_0:th.27">Th27</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="#E2:79"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">N</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c2">N</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font></span><br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">N</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c2">N</font>) = 
<font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c2">N</font>)

<br/>.= 
<font color="Maroon" title="c4">a'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">M</font>)
<b>by </b><i><a class="txt" href="#E1:79"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E2:79"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="monoid_0.html#T27" target="_self" title="MONOID_0:th.27">Th27</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="#E2:79"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:79"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c2">N</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="c2">N</font>)
 <b>by </b><i><a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D21" target="_self" title="MONOID_0:def.21">MONOID_0:def 21</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
