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<b>let </b><font color="Maroon">it1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="fintopo2.html#V1" target="_self">strict</a>  <a href="fintopo2.html#L1" target="_self">FMT_Space_Str</a> , <font color="Maroon">it2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="fintopo2.html#V1" target="_self">strict</a>  <a href="fintopo2.html#L1" target="_self">FMT_Space_Str</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:22_1_2"/><i><font color="Green">E35</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E2:22_1_2"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">it1</font> holds   <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E3:22_1_2"/><i><font color="Green">E37</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">it2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E4:22_1_2"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">it2</font> holds   <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) </span>}</span> 
 ;<br/>

<a NAME="E5:22_1_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font> holds  the <a href="fintopo2.html#U1" target="_self">BNbd</a> of <font color="Maroon">it1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font>;<br/>

<a NAME="E1:22_1_2_1"/>
the <a href="fintopo2.html#U1" target="_self">BNbd</a> of <font color="Maroon">it1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x</font>
 
;<br/>
<b>hence </b><a NAME="E2:22_1_2_1"/>
the <a href="fintopo2.html#U1" target="_self">BNbd</a> of <font color="Maroon">it1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:22_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font> holds  the <a href="fintopo2.html#U1" target="_self">BNbd</a> of <font color="Maroon">it2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font>;<br/>

<a NAME="E1:22_1_2_2"/>
the <a href="fintopo2.html#U1" target="_self">BNbd</a> of <font color="Maroon">it2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="fintopo2.html#K6" target="_self">U_FMT</a> <font color="Maroon">x</font>
 
;<br/>
<b>hence </b><a NAME="E2:22_1_2_2"/>
the <a href="fintopo2.html#U1" target="_self">BNbd</a> of <font color="Maroon">it2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">it1</font>;<br/><b>thus </b>the <a href="fintopo2.html#U1" target="_self">BNbd</a> of <font color="Maroon">it1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) </span>}</span> 
<b>by </b><i/>
<br/>.= 
the <a href="fintopo2.html#U1" target="_self">BNbd</a> of <font color="Maroon">it2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:22_1_2"/>
<font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">it2</font>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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