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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<a NAME="E2:13"/><i><font color="Green">E13</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<a NAME="E3:13"/><i><font color="Green">E14</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:13_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<a NAME="E3:13_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<b>hence </b><a NAME="E4:13_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K5">NAT</a> </span>)</span>
 <b>by </b><i><a class="txt" href="frechet.html#E2:13"><i><font color="Green">E13</font></i></a>, <a class="txt" href="frechet.html#E2:13_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="partfun1.html#K6">id</a> <a href="numbers.html#K5">NAT</a>  as   <a href="normsp_1.html#NM2">sequence</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="frechet.html#E2:13"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E5:13"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<a NAME="E2:13_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<a NAME="E3:13_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E4:13_2"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E5:13_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="frechet.html#E2:13_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="frechet.html#E3:13_2"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:13_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:13"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="frechet.html#E1:13"><i><font color="Green">E12</font></i></a>, <a class="txt" href="frechet.html#E3:13"><i><font color="Green">E14</font></i></a>, <a class="ref" href="frechet.html#T9" target="_self">Th9</a></i>;<br/>


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