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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</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   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a><br/>  for <font color="Olive" title="b2">T</font> being   <a href="c0sp3.html#NM1" title="C0SP3:NM.1">NormedLinearTopSpace</a>  st <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Olive" title="b2">T</font> is  <a href="lopban_1.html#V3" title="LOPBAN_1:attr.3">complete</a>  &amp; <font color="Olive" title="b2">T</font> is  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a>  &amp;  <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Olive" title="b2">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> for <font color="Olive" title="b3">G</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">T</font>)</span>)</span><br/>  for <font color="Olive" title="b4">F</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="c0sp3.html#K9" title="C0SP3:func.9">R_NormSpace_of_ContinuousFunctions</a> (<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">T</font>)</span>)</span>  st <font color="Olive" title="b3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">F</font> holds <br/>(  <a href="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Olive" title="b4">F</font> is  <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a>  iff ( <font color="Olive" title="b3">G</font> is  <a href="ascoli.html#V1" title="ASCOLI:attr.1">equibounded</a>  &amp; <font color="Olive" title="b3">G</font> is  <a href="ascoli.html#V2" title="ASCOLI:attr.2">equicontinuous</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</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">T</font> being   <a href="c0sp3.html#NM1" title="C0SP3:NM.1">NormedLinearTopSpace</a>  st <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Olive" title="b1">T</font> is  <a href="lopban_1.html#V3" title="LOPBAN_1:attr.3">complete</a>  &amp; <font color="Olive" title="b1">T</font> is  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a>  &amp;  <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Olive" title="b1">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> for <font color="Olive" title="b2">G</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">T</font>)</span>)</span><br/>  for <font color="Olive" title="b3">F</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="c0sp3.html#K9" title="C0SP3:func.9">R_NormSpace_of_ContinuousFunctions</a> (<font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">T</font>)</span>)</span>  st <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F</font> holds <br/>(  <a href="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Olive" title="b3">F</font> is  <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a>  iff ( <font color="Olive" title="b2">G</font> is  <a href="ascoli.html#V1" title="ASCOLI:attr.1">equibounded</a>  &amp; <font color="Olive" title="b2">G</font> is  <a href="ascoli.html#V2" title="ASCOLI:attr.2">equicontinuous</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">T</font> be   <a href="c0sp3.html#NM1" title="C0SP3:NM.1">NormedLinearTopSpace</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"> ( <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c3">T</font> is  <a href="lopban_1.html#V3" title="LOPBAN_1:attr.3">complete</a>  &amp; <font color="Maroon" title="c3">T</font> is  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a>  &amp;  <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c3">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  for <font color="Olive" title="b1">G</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font>)</span>)</span><br/>  for <font color="Olive" title="b2">F</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="c0sp3.html#K9" title="C0SP3:func.9">R_NormSpace_of_ContinuousFunctions</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>)</span>)</span>  st <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> holds <br/>(  <a href="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Olive" title="b2">F</font> is  <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a>  iff ( <font color="Olive" title="b1">G</font> is  <a href="ascoli.html#V1" title="ASCOLI:attr.1">equibounded</a>  &amp; <font color="Olive" title="b1">G</font> is  <a href="ascoli.html#V2" title="ASCOLI:attr.2">equicontinuous</a>  ) ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">and </span><br/><a NAME="E2:19"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c3">T</font> is  <a href="lopban_1.html#V3" title="LOPBAN_1:attr.3">complete</a> 
 <span class="kw">and </span><br/><a NAME="E3:19"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
( <font color="Maroon" title="c3">T</font> is  <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a>  &amp;  <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c3">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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">G</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font>)</span>)</span><br/>  for <font color="Olive" title="b2">F</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="c0sp3.html#K9" title="C0SP3:func.9">R_NormSpace_of_ContinuousFunctions</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>)</span>)</span>  st <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> holds <br/>(  <a href="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Olive" title="b2">F</font> is  <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a>  iff ( <font color="Olive" title="b1">G</font> is  <a href="ascoli.html#V1" title="ASCOLI:attr.1">equibounded</a>  &amp; <font color="Olive" title="b1">G</font> is  <a href="ascoli.html#V2" title="ASCOLI:attr.2">equicontinuous</a>  ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">G</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</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">F</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="c0sp3.html#K9" title="C0SP3:func.9">R_NormSpace_of_ContinuousFunctions</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>)</span>)</span>  st <font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> holds <br/>(  <a href="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Olive" title="b1">F</font> is  <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a>  iff ( <font color="Maroon" title="c4">G</font> is  <a href="ascoli.html#V1" title="ASCOLI:attr.1">equibounded</a>  &amp; <font color="Maroon" title="c4">G</font> is  <a href="ascoli.html#V2" title="ASCOLI:attr.2">equicontinuous</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">F</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="c0sp3.html#K9" title="C0SP3:func.9">R_NormSpace_of_ContinuousFunctions</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</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="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> implies (  <a href="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Maroon" title="c5">F</font> is  <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a>  iff ( <font color="Maroon" title="c4">G</font> is  <a href="ascoli.html#V1" title="ASCOLI:attr.1">equibounded</a>  &amp; <font color="Maroon" title="c4">G</font> is  <a href="ascoli.html#V2" title="ASCOLI:attr.2">equicontinuous</a>  ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E4:19"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font>
 ; <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="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Maroon" title="c5">F</font> is  <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a>  iff ( <font color="Maroon" title="c4">G</font> is  <a href="ascoli.html#V1" title="ASCOLI:attr.1">equibounded</a>  &amp; <font color="Maroon" title="c4">G</font> is  <a href="ascoli.html#V2" title="ASCOLI:attr.2">equicontinuous</a>  ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">H</font> = <font color="Maroon" title="c5">F</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <span class="p2">(<span class="default"><a href="c0sp3.html#K9" title="C0SP3:func.9">R_NormSpace_of_ContinuousFunctions</a> (<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>)</span>)</span></span>)</span> ;<br/>
<a NAME="E6:19"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 <a href="ascoli.html#K1" title="ASCOLI:func.1">Cl</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ascoli.html#T1" onmouseover="rs('ascoli/T1')" onmouseout="rh()">ASCOLI:1</a></span>;<br/>
<a NAME="E7:19"/>
(  <a href="ascoli.html#K1" title="ASCOLI:func.1">Cl</a> <font color="Maroon" title="c6">H</font> is  <a href="topmetr4.html#V1" title="TOPMETR4:attr.1">sequentially_compact</a>  iff ( <font color="Maroon" title="c4">G</font> is  <a href="ascoli.html#V1" title="ASCOLI:attr.1">equibounded</a>  &amp; <font color="Maroon" title="c4">G</font> is  <a href="ascoli.html#V2" title="ASCOLI:attr.2">equicontinuous</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="neurons1.html#T27" target="_self" onmouseover="rs('neurons1/T27')" onmouseout="rh()">Th27</a>, <a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:19"/>
(  <a href="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Maroon" title="c5">F</font> is  <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a>  iff ( <font color="Maroon" title="c4">G</font> is  <a href="ascoli.html#V1" title="ASCOLI:attr.1">equibounded</a>  &amp; <font color="Maroon" title="c4">G</font> is  <a href="ascoli.html#V2" title="ASCOLI:attr.2">equicontinuous</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr4.html#T18" onmouseover="rs('topmetr4/T18')" onmouseout="rh()">TOPMETR4:18</a>, <a class="txt" href="#E6:19"><span class="lab"><font color="Green" title="E20">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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