<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="lopban_1.html#NM2" title="LOPBAN_1:NM.2">RealBanachSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">X0</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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>;<br/><b>let </b><font color="Maroon" title="c4">vseq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c3">X0</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 ;<br/>

<b>assume </b><a NAME="E2:11"/><i><font color="Green" title="E9">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X0</font> holds <br/><font color="Maroon" title="c4">vseq</font> <a href="lopban_5.html#K2" title="LOPBAN_5:func.2">#</a> <font color="Olive" title="b1">x</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 ;<br/>

<b>assume </b><a NAME="E3:11"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">K</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">K</font> &amp; ( for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">vseq</font> <a href="lopban_5.html#K2" title="LOPBAN_5:func.2">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">K</font> ) )
 ;<br/>

<a NAME="E4:11"/><b>then </b><i><font color="Green" title="E10">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c4">vseq</font> <a href="lopban_5.html#K2" title="LOPBAN_5:func.2">#</a> <font color="Olive" title="b1">x</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="lopban_5.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="lopban_5.html#E2:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="lopban_5.html#T7" target="_self" title="LOPBAN_5:th.7">Th7</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> =  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">vseq</font> <a href="lopban_5.html#K2" title="LOPBAN_5:func.2">#</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c5">tseq</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E6:11"/><i><font color="Green" title="E11">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c5">tseq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c6">tseq</font> = <font color="Maroon" title="c5">tseq</font> as  <a href="lopban_1.html#V3" title="LOPBAN_1:attr.3">bounded</a> <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="txt" href="lopban_5.html#E4:11"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="lopban_5.html#E6:11"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="lopban_5.html#T6" target="_self" title="LOPBAN_5:th.6">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">tseq</font> = <font color="Maroon" title="c6">tseq</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <b>by </b><i><a class="ref" href="lopban_1.html#D10" title="LOPBAN_1:def.10">LOPBAN_1:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">tseq</font>
;<br/>

<b>thus </b><a NAME="E9:11"/>
( ( for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Maroon" title="c4">vseq</font> <a href="lopban_5.html#K2" title="LOPBAN_5:func.2">#</a> <font color="Olive" title="b1">x</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c7">tseq</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">vseq</font> <a href="lopban_5.html#K2" title="LOPBAN_5:func.2">#</a> <font color="Olive" title="b1">x</font></span>)</span> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">tseq</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">vseq</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> ) ) &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c7">tseq</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">vseq</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> )
 <b>by </b><i><a class="txt" href="lopban_5.html#E4:11"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="lopban_5.html#E6:11"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="lopban_5.html#T6" target="_self" title="LOPBAN_5:th.6">Th6</a></i>;<br/>


</div>
