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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
( <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>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D21" target="_self" title="MONOID_0:def.21">Def21</a></span>;<br/>
<span class="kw">hence </span><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> ) )
 <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E3:37"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><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">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>;<br/><span class="kw">hence </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="binop_1.html#T3" title="BINOP_1:th.3">BINOP_1:3</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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