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<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="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"> ( <font color="Maroon" title="c1">M</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> iff for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds <br/> ( <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> ) )</span><br/>

<a NAME="E1:37"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c1">M</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> iff <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></i>;<br/>
<b>hence </b><a NAME="E2:37"/>
( <font color="Maroon" title="c1">M</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> implies for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds <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="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> ) )
 <b>by </b><i><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"> ( ( for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds <br/> ( <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> ) ) implies <font color="Maroon" title="c1">M</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> )</span><br/>

<b>assume </b><a NAME="E3:37"/><i><font color="Green" title="E26">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds <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="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">M</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">H<sub>2</sub></font>(<font color="Maroon" title="c1">M</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="c1">M</font>),<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">M</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )</span><br/><a NAME="E1:37_1"/>
( <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="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">H<sub>3</sub></font>(<font color="Maroon" title="c1">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E3:37"><i><font color="Green" title="E26">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E2:37_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">M</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="c1">M</font>),<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">M</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 ; <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="E5:37"/>
<font color="Maroon" title="c1">M</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:37"><i><font color="Green" title="E25">A1</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/>


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