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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span><br/>  for <font color="Olive" title="b2">St</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Olive" title="b1">S</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">St</font> &amp; <font color="Olive" title="b2">St</font> is  <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a>  holds <br/>(  <a href="frechet.html#K2" title="FRECHET:func.2">Lim</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet.html#K2" title="FRECHET:func.2">Lim</a> <font color="Olive" title="b2">St</font> &amp;  <a href="frechet2.html#K2" title="FRECHET2:func.2">lim</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K2" title="FRECHET2:func.2">lim</a> <font color="Olive" title="b2">St</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">St</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Maroon" title="c2">S</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">St</font> &amp; <font color="Olive" title="b1">St</font> is  <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a>  holds <br/>(  <a href="frechet.html#K2" title="FRECHET:func.2">Lim</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet.html#K2" title="FRECHET:func.2">Lim</a> <font color="Olive" title="b1">St</font> &amp;  <a href="frechet2.html#K2" title="FRECHET2:func.2">lim</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K2" title="FRECHET2:func.2">lim</a> <font color="Olive" title="b1">St</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">St</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">St</font> &amp; <font color="Maroon" title="c3">St</font> is  <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a>  implies (  <a href="frechet.html#K2" title="FRECHET:func.2">Lim</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet.html#K2" title="FRECHET:func.2">Lim</a> <font color="Maroon" title="c3">St</font> &amp;  <a href="frechet2.html#K2" title="FRECHET2:func.2">lim</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K2" title="FRECHET2:func.2">lim</a> <font color="Maroon" title="c3">St</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">St</font>
 <span class="kw">and </span><br/><a NAME="E2:39"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c3">St</font> is  <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="frechet.html#K2" title="FRECHET:func.2">Lim</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet.html#K2" title="FRECHET:func.2">Lim</a> <font color="Maroon" title="c3">St</font> &amp;  <a href="frechet2.html#K2" title="FRECHET2:func.2">lim</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K2" title="FRECHET2:func.2">lim</a> <font color="Maroon" title="c3">St</font> )</span><br/>

<a NAME="E3:39"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c2">S</font> is  <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="normsp_2.html#T27" target="_self" onmouseover="rs('normsp_2/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:39"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="frechet.html#D4" onmouseover="rs('frechet/D4')" onmouseout="rh()">FRECHET:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">xxt</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" target="_self" onmouseover="rs('normsp_2/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E7:39"/>
<font color="Maroon" title="c3">St</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c5">xxt</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="normsp_2.html#T26" target="_self" onmouseover="rs('normsp_2/T26')" onmouseout="rh()">Th26</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E32">A5</font></span>:  <a href="frechet2.html#K2" title="FRECHET2:func.2">lim</a> <font color="Maroon" title="c3">St</font> = 
<font color="Maroon" title="c5">xxt</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="frechet2.html#T25" onmouseover="rs('frechet2/T25')" onmouseout="rh()">FRECHET2:25</a></span>
<br/>.= 
 <a href="frechet2.html#K2" title="FRECHET2:func.2">lim</a> <font color="Maroon" title="c2">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="frechet2.html#T25" onmouseover="rs('frechet2/T25')" onmouseout="rh()">FRECHET2:25</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">Sn</font> = <font color="Maroon" title="c2">S</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">xt</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:39"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
 <a href="frechet.html#K2" title="FRECHET:func.2">Lim</a> <font color="Maroon" title="c3">St</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">xt</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="frechet2.html#T24" onmouseover="rs('frechet2/T24')" onmouseout="rh()">FRECHET2:24</a></span>;<br/>
<a NAME="E12:39"/>
<font color="Maroon" title="c7">xt</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">xt</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E13:39"/><span class="kw">then </span>
<font color="Maroon" title="c3">St</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c7">xt</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="frechet.html#D5" onmouseover="rs('frechet/D5')" onmouseout="rh()">FRECHET:def 5</a></span>;<br/>
<span class="kw">then </span> <a href="frechet.html#K2" title="FRECHET:func.2">Lim</a> <font color="Maroon" title="c3">St</font> = 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="frechet2.html#K2" title="FRECHET2:func.2">lim</a> <font color="Maroon" title="c3">St</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="frechet2.html#T25" onmouseover="rs('frechet2/T25')" onmouseout="rh()">FRECHET2:25</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <font color="Maroon" title="c6">Sn</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="normsp_2.html#T14" target="_self" onmouseover="rs('normsp_2/T14')" onmouseout="rh()">Th14</a></span>
<br/>.= 
 <a href="frechet.html#K2" title="FRECHET:func.2">Lim</a> <font color="Maroon" title="c2">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="normsp_2.html#T14" target="_self" onmouseover="rs('normsp_2/T14')" onmouseout="rh()">Th14</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:39"/>
(  <a href="frechet.html#K2" title="FRECHET:func.2">Lim</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet.html#K2" title="FRECHET:func.2">Lim</a> <font color="Maroon" title="c3">St</font> &amp;  <a href="frechet2.html#K2" title="FRECHET2:func.2">lim</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K2" title="FRECHET2:func.2">lim</a> <font color="Maroon" title="c3">St</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E32">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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