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

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M1</font>, <font color="Olive" title="b2">M2</font>, <font color="Olive" title="b3">M3</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Olive" title="b2">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Olive" title="b3">M3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Olive" title="b3">M3</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c3">M2</font>, <font color="Maroon" title="c4">M3</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span></span><br/>



<a NAME="E1:83"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E3:83_1"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">x9</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> )
 <span class="kw">and </span><br/><a NAME="E4:83_1"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
<font color="Maroon" title="c7">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">x2</font>, <font color="Maroon" title="c9">x3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E6:83_1"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
( <font color="Maroon" title="c7">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">x3</font> &amp; <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M2</font> )
 <span class="kw">and </span><br/><a NAME="E7:83_1"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
<font color="Maroon" title="c9">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_1"><span class="lab"><font color="Green" title="E49">A2</font></span></a></span>;<br/>
<a NAME="E8:83_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">x2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">x3</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E6:83_1"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:83_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_1"><span class="lab"><font color="Green" title="E51">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></span>;<br/>
<a NAME="E3:83"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x9</font>, <font color="Maroon" title="c7">x3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E3:83_2"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x9</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">x3</font>
 <span class="kw">and </span><br/><a NAME="E4:83_2"/><span class="lab"><font color="Green" title="E50">A7</font></span>: 
<font color="Maroon" title="c6">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font>
 <span class="kw">and </span><br/><a NAME="E5:83_2"/><span class="lab"><font color="Green" title="E51">A8</font></span>: 
<font color="Maroon" title="c7">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M3</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E7:83_2"/><span class="lab"><font color="Green" title="E52">A9</font></span>: 
<font color="Maroon" title="c6">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">x2</font>
 <span class="kw">and </span><br/><a NAME="E8:83_2"/><span class="lab"><font color="Green" title="E53">A10</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font>
 <span class="kw">and </span><br/><a NAME="E9:83_2"/><span class="lab"><font color="Green" title="E54">A11</font></span>: 
<font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_2"><span class="lab"><font color="Green" title="E50">A7</font></span></a></span>;<br/>
<a NAME="E10:83_2"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">x3</font></span>)</span> &amp; <font color="Maroon" title="c9">x2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_2"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E5:83_2"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="txt" href="#E7:83_2"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E9:83_2"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:83_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_2"><span class="lab"><font color="Green" title="E53">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:83"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:83"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">M2</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M1</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c4">M3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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