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<div class="add">

<b>let </b><font color="Maroon" title="c1">X</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> ;<br/><b>let </b><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:6"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c3">N</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> </span>}</span> 
 ;<br/>

<b>thus </b><a NAME="E2:6"/>
<font color="Maroon" title="c2">M</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">N</font>
 ;<br/>

<a NAME="E2:6_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font></span>)</span> where <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10" title="RUSUB_4:def.10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">u</font>, <font color="Maroon" title="c7">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:6_1"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">v</font>
 <b>and </b><br/><a NAME="E5:6_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <b>and </b><br/><a NAME="E6:6_1"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 ;<br/>
<a NAME="E7:6_1"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c3">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E1:6"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="rltopsp1.html#E5:6_1"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>
<a NAME="E8:6_1"/>
<font color="Maroon" title="c6">u</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">v'</font></span>)</span> where <font color="Olive" title="b1">v'</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D9" title="RUSUB_4:def.9">RUSUB_4:def 9</a></i>;<br/>
<a NAME="E9:6_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">u</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E4:6_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="rltopsp1.html#E6:6_1"><i><font color="Green" title="E9">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:6_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E7:6_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:6"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:6"/><i><font color="Green" title="E7">A6</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 <b>and </b><br/><a NAME="E6:6"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E8:6"/><i><font color="Green" title="E9">A8</font></i>: 
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c3">N</font>
 <b>and </b><br/><a NAME="E9:6"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:6"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="rltopsp1.html#E6:6"><i><font color="Green" title="E8">A7</font></i></a></i>;<br/>
<a NAME="E10:6"/>
<font color="Maroon" title="c7">u</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c7">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">v</font></span>)</span> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D9" title="RUSUB_4:def.9">RUSUB_4:def 9</a></i>;<br/>
<a NAME="E11:6"/><b>then </b>
 ex <font color="Olive" title="b1">v</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">v</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> )
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E5:6"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="rltopsp1.html#E8:6"><i><font color="Green" title="E9">A8</font></i></a></i>;<br/>
<a NAME="E12:6"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">u'</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v'</font></span>)</span> where <font color="Olive" title="b1">u'</font>, <font color="Olive" title="b2">v'</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b2">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E9:6"><i><font color="Green" title="E10">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:6"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="ref" href="rusub_4.html#D10" title="RUSUB_4:def.10">RUSUB_4:def 10</a></i>;<br/>


</div>
