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<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of $1, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of $1</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of $1</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> =  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of $1;<br/>
<span class="kw">let </span><font color="Maroon" title="c1">LS</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#L2" title="ALGSTR_0:struct.2">addLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">LS</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp;  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">LS</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  implies ( <font color="Maroon" title="c1">LS</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp; <font color="Maroon" title="c1">LS</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:18"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">LS</font>) is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> 
 <span class="kw">and </span><br/><a NAME="E2:18"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">LS</font>) is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</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">LS</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp; <font color="Maroon" title="c1">LS</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  )</span><br/>

<a NAME="E3:18"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">LS</font> holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:18"/>
<font color="Maroon" title="c1">LS</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D2" title="RLVECT_1:def.2">RLVECT_1:def 2</a></span>; <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">LS</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> </span><br/>

<a NAME="E5:18"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">LS</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:18"/>
<font color="Maroon" title="c1">LS</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 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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