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<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">St</font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/><b>let </b><font color="Maroon">xt</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:39"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">S</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">St</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xt</font> )
 ;<br/>

<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">St</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">xt</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">U1</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/><b>assume </b><a NAME="E1:39_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">U1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font> )
 ;<br/><b>reconsider </b><font color="Maroon">U1t</font> = <font color="Maroon">U1</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:39_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">St</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1t</font>
 <b>by </b><i><a class="ref" href="frechet.html#D2">FRECHET:def 2</a>, , , </i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E5:39_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><b>hence </b><a NAME="E6:39_1_1"/>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:39_1"/>
<font color="Maroon">S</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="frechet.html#D2">FRECHET:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">U1t</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/><b>assume </b><a NAME="E1:39_2_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">U1t</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">xt</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1t</font> )
 ;<br/><b>reconsider </b><font color="Maroon">U1</font> = <font color="Maroon">U1t</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:39_2_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 <b>by </b><i><a class="ref" href="frechet.html#D2">FRECHET:def 2</a>, , , </i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E5:39_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><b>hence </b><a NAME="E6:39_2_1"/>
<font color="Maroon">St</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1t</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:39_2"/>
<font color="Maroon">St</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">xt</font>
 <b>by </b><i><a class="ref" href="frechet.html#D2">FRECHET:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39"/>
( <font color="Maroon">St</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">xt</font> iff <font color="Maroon">S</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/>


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