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

<span class="kw">let </span><font color="Maroon" title="c1">UN</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">UN</font> &amp;  <a href="grcat_1.html#R1" title="GRCAT_1:pred.1">GO</a> <font color="Olive" title="b1">x</font>, <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">x2</font> =  <a href="funct_5.html#K9" title="FUNCT_5:func.9">op2</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">UN</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#T6" target="_self" title="GRCAT_1:th.6">Th6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">x3</font> =  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">UN</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#T6" target="_self" title="GRCAT_1:th.6">Th6</a>, <a class="ref" href="grcat_1.html#T13" target="_self" title="GRCAT_1:th.13">Th13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">u</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">UN</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="classes2.html#T62" title="CLASSES2:th.62">CLASSES2:62</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">x1</font> = <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>;<br/>
<a NAME="E4:86"/>
 <a href="algstr_1.html#K1" title="ALGSTR_1:func.1">Extract</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">x4</font> =  <a href="algstr_1.html#K1" title="ALGSTR_1:func.1">Extract</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">UN</font> ;<br/>
<span class="kw">take </span><font color="Maroon" title="c7">x</font> = <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">UN</font> &amp;  <a href="grcat_1.html#R1" title="GRCAT_1:pred.1">GO</a> <font color="Maroon" title="c7">x</font>, <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E6:86"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">UN</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#T3" target="_self" title="GRCAT_1:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="grcat_1.html#R1" title="GRCAT_1:pred.1">GO</a> <font color="Maroon" title="c7">x</font>, <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a> </span><br/>

<span class="kw">take </span>
<span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="grcat_1.html#D28" target="_self" title="GRCAT_1:def.28">GRCAT_1:def 28</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x2</font>, <font color="Olive" title="b2">x3</font>, <font color="Olive" title="b3">x4</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>,<font color="Olive" title="b1">x2</font>,<font color="Olive" title="b2">x3</font>,<font color="Olive" title="b3">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> &amp;  ex <font color="Olive" title="b4">G</font> being  <a href="algstr_0.html#V8" title="ALGSTR_0:attr.8">strict</a>  <a href="vectsp_1.html#NM1" title="VECTSP_1:NM.1">AddGroup</a> st <br/>(  <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">G</font> &amp; <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b4">G</font> &amp; <font color="Olive" title="b1">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b4">G</font> &amp; <font color="Olive" title="b2">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b4">G</font> &amp; <font color="Olive" title="b3">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b4">G</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c2">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x3</font>, <font color="Olive" title="b2">x4</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>,<font color="Maroon" title="c2">x2</font>,<font color="Olive" title="b1">x3</font>,<font color="Olive" title="b2">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> &amp;  ex <font color="Olive" title="b3">G</font> being  <a href="algstr_0.html#V8" title="ALGSTR_0:attr.8">strict</a>  <a href="vectsp_1.html#NM1" title="VECTSP_1:NM.1">AddGroup</a> st <br/>(  <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">G</font> &amp; <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">G</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b3">G</font> &amp; <font color="Olive" title="b1">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b3">G</font> &amp; <font color="Olive" title="b2">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b3">G</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c3">x3</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x4</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font>,<font color="Olive" title="b1">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> &amp;  ex <font color="Olive" title="b2">G</font> being  <a href="algstr_0.html#V8" title="ALGSTR_0:attr.8">strict</a>  <a href="vectsp_1.html#NM1" title="VECTSP_1:NM.1">AddGroup</a> st <br/>(  <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G</font> &amp; <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b2">G</font> &amp; <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b1">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b2">G</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">x4</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="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> &amp;  ex <font color="Olive" title="b1">G</font> being  <a href="algstr_0.html#V8" title="ALGSTR_0:attr.8">strict</a>  <a href="vectsp_1.html#NM1" title="VECTSP_1:NM.1">AddGroup</a> st <br/>(  <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp; <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c6">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">G</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:86"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font>,<font color="Maroon" title="c6">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being  <a href="algstr_0.html#V8" title="ALGSTR_0:attr.8">strict</a>  <a href="vectsp_1.html#NM1" title="VECTSP_1:NM.1">AddGroup</a> st <br/>(  <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp; <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c6">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">G</font> )</span><br/>

<span class="kw">take </span>
 <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  &amp; <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  &amp; <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  &amp; <font color="Maroon" title="c6">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E8:86"/>
(  <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  &amp; <span class="p1"><a href="classes2.html#K1" title="CLASSES2:func.1">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="classes2.html#K1" title="CLASSES2:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  &amp; <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  &amp; <font color="Maroon" title="c6">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</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>
