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

<span class="kw">let </span><font color="Maroon" title="c1">K</font>, <font color="Maroon" title="c2">L</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">ModuleStr</a> over <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b2">W</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="vectsp_2.html#L1" title="VECTSP_2:struct.1">RightModStr</a> over <font color="Maroon" title="c2">L</font><br/>  for <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b5">w1</font>, <font color="Olive" title="b6">w2</font> being   <a href="vectsp_2.html#NM4" title="VECTSP_2:NM.4">Vector</a> of <font color="Olive" title="b2">W</font>  st <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K4" title="MOD_4:func.4">opp</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">w1</font> &amp; <font color="Olive" title="b4">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">w2</font> holds <br/><font color="Olive" title="b5">w1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b6">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b4">v2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">V</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">ModuleStr</a> over <font color="Maroon" title="c1">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="vectsp_2.html#L1" title="VECTSP_2:struct.1">RightModStr</a> over <font color="Maroon" title="c2">L</font><br/>  for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">V</font><br/>  for <font color="Olive" title="b4">w1</font>, <font color="Olive" title="b5">w2</font> being   <a href="vectsp_2.html#NM4" title="VECTSP_2:NM.4">Vector</a> of <font color="Olive" title="b1">W</font>  st <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K4" title="MOD_4:func.4">opp</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w1</font> &amp; <font color="Olive" title="b3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">w2</font> holds <br/><font color="Olive" title="b4">w1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b5">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">v2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">W</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="vectsp_2.html#L1" title="VECTSP_2:struct.1">RightModStr</a> over <font color="Maroon" title="c2">L</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">V</font><br/>  for <font color="Olive" title="b3">w1</font>, <font color="Olive" title="b4">w2</font> being   <a href="vectsp_2.html#NM4" title="VECTSP_2:NM.4">Vector</a> of <font color="Maroon" title="c4">W</font>  st <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K4" title="MOD_4:func.4">opp</a> <font color="Maroon" title="c4">W</font> &amp; <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w1</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w2</font> holds <br/><font color="Olive" title="b3">w1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b4">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v2</font></span><br/>





<a NAME="E1:34"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K4" title="MOD_4:func.4">opp</a> <font color="Maroon" title="c4">W</font></span>)</span>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K4" title="MOD_4:func.4">opp</a> <font color="Maroon" title="c4">W</font></span>)</span>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K4" title="MOD_4:func.4">opp</a> <font color="Maroon" title="c4">W</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c4">W</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c4">W</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T9" target="_self" onmouseover="rs('mod_4/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">v1</font>, <font color="Maroon" title="c6">v2</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">w1</font>, <font color="Olive" title="b2">w2</font> being   <a href="vectsp_2.html#NM4" title="VECTSP_2:NM.4">Vector</a> of <font color="Maroon" title="c4">W</font>  st <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K4" title="MOD_4:func.4">opp</a> <font color="Maroon" title="c4">W</font> &amp; <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w2</font> holds <br/><font color="Olive" title="b1">w1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">v2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">w1</font>, <font color="Maroon" title="c8">w2</font> be   <a href="vectsp_2.html#NM4" title="VECTSP_2:NM.4">Vector</a> of <font color="Maroon" title="c4">W</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K4" title="MOD_4:func.4">opp</a> <font color="Maroon" title="c4">W</font> &amp; <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">w1</font> &amp; <font color="Maroon" title="c6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w2</font> implies <font color="Maroon" title="c7">w1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">v2</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:34"/>
( <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mod_4.html#K4" title="MOD_4:func.4">opp</a> <font color="Maroon" title="c4">W</font> &amp; <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">w1</font> &amp; <font color="Maroon" title="c6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">w1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">v2</font></span><br/>

<span class="kw">hence </span><a NAME="E3:34"/>
<font color="Maroon" title="c7">w1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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