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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">H</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="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="Olive" title="b1">S</font>,<font color="Olive" title="b2">T</font>)</span>)</span></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">H</font> holds <br/>( <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="Olive" title="b1">S</font>,<font color="Olive" title="b2">T</font>)</span>)</span></span>)</span> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Olive" title="b4">H</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</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">H</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="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="Olive" title="b1">T</font>)</span>)</span></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">H</font> holds <br/>( <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="Olive" title="b1">T</font>)</span>)</span></span>)</span> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Olive" title="b3">H</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</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">H</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="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>  st <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H</font> holds <br/>( <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> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Olive" title="b2">H</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</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="E1:17"/><span class="lab"><font color="Green" title="E14">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="E2:17"/><span class="lab"><font color="Green" title="E15">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="E3:17"/><span class="lab"><font color="Green" title="E16">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">H</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="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>  st <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H</font> holds <br/>( <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> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Olive" title="b2">H</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</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">H</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="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>  st <font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H</font> holds <br/>( <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> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Olive" title="b1">H</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</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">H</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="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>; <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">H</font> implies ( <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> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c5">H</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</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="E4:17"/><span class="lab"><font color="Green" title="E17">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">H</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"> ( <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> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c5">H</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</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">hence </span><a NAME="E5:17"/>
( <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> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c5">H</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</a>  implies ( <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="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="ascoli.html#T13" onmouseover="rs('ascoli/T13')" onmouseout="rh()">ASCOLI:13</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"> ( <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>  implies <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> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c5">H</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E6:17"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
( <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>  )
 ; <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"> <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> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c5">H</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</a> </span><br/>

<a NAME="E7:17"/>
 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="c2">S</font><br/>  for <font color="Olive" title="b2">Hx</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="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c3">T</font></span>)</span>  st <font color="Olive" title="b2">Hx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b3">f</font> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font> : <font color="Olive" title="b3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">H</font> </span> } </span>  holds <br/><span class="p1">(<span class="default"><a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <span class="p1">(<span class="default"><a href="ascoli.html#K1" title="ASCOLI:func.1">Cl</a> <font color="Olive" title="b2">Hx</font></span>)</span> is  <a href="topmetr.html#V2" title="TOPMETR:attr.2">compact</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</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">  for <font color="Olive" title="b1">Hx</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="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c3">T</font></span>)</span>  st <font color="Olive" title="b1">Hx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> where <font color="Olive" title="b2">f</font> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font> : <font color="Olive" title="b2">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">H</font> </span> } </span>  holds <br/><span class="p1">(<span class="default"><a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <span class="p1">(<span class="default"><a href="ascoli.html#K1" title="ASCOLI:func.1">Cl</a> <font color="Olive" title="b1">Hx</font></span>)</span> is  <a href="topmetr.html#V2" title="TOPMETR:attr.2">compact</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c7">Hx</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="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <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="c7">Hx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font> : <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">H</font> </span> } </span>  implies <span class="p1">(<span class="default"><a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <span class="p1">(<span class="default"><a href="ascoli.html#K1" title="ASCOLI:func.1">Cl</a> <font color="Maroon" title="c7">Hx</font></span>)</span> is  <a href="topmetr.html#V2" title="TOPMETR:attr.2">compact</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:17_1"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<font color="Maroon" title="c7">Hx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font> : <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">H</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"> <span class="p1">(<span class="default"><a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <span class="p1">(<span class="default"><a href="ascoli.html#K1" title="ASCOLI:func.1">Cl</a> <font color="Maroon" title="c7">Hx</font></span>)</span> is  <a href="topmetr.html#V2" title="TOPMETR:attr.2">compact</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">Tx</font> = <font color="Maroon" title="c7">Hx</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#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c3">T</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">Fx</font> = <font color="Maroon" title="c7">Hx</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 <font color="Maroon" title="c3">T</font> ;<br/>
<a NAME="E4:17_1"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
 <a href="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Maroon" title="c9">Fx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ascoli.html#K1" title="ASCOLI:func.1">Cl</a> <font color="Maroon" title="c7">Hx</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/>
<span class="kw">consider </span><font color="Maroon" title="c10">RNS</font> being   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><span class="kw"> such that </span><br/><a NAME="E6:17_1"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
( <font color="Maroon" title="c10">RNS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#G1" title="NORMSP_1:aggr.1">NORMSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c3">T</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">T</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c3">T</font>, the <a href="normsp_0.html#U1" title="NORMSP_0:sel.1">normF</a> of <font color="Maroon" title="c3">T</font> #) &amp;  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</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="c10">RNS</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp3.html#D6" onmouseover="rs('c0sp3/D6')" onmouseout="rh()">C0SP3:def 6</a></span>;<br/>
<a NAME="E7:17_1"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
( <font color="Maroon" title="c10">RNS</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="c10">RNS</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E6:17_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="neurons1.html#T25" target="_self" onmouseover="rs('neurons1/T25')" onmouseout="rh()">Th25</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">Fx0</font> = <font color="Maroon" title="c9">Fx</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 <font color="Maroon" title="c10">RNS</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a></span>;<br/>
<a NAME="E9:17_1"/><span class="lab"><font color="Green" title="E23">A10</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">RNS</font> holds  <span class="p1">(<span class="default"><a href="normsp_2.html#K1" title="NORMSP_2:func.1">distance_by_norm_of</a> <font color="Maroon" title="c10">RNS</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="normsp_2.html#K1" title="NORMSP_2:func.1">distance_by_norm_of</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">RNS</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"> <span class="p1">(<span class="default"><a href="normsp_2.html#K1" title="NORMSP_2:func.1">distance_by_norm_of</a> <font color="Maroon" title="c10">RNS</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="normsp_2.html#K1" title="NORMSP_2:func.1">distance_by_norm_of</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font>)</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c14">a1</font> = <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c15">b1</font> = <font color="Maroon" title="c13">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E24">A11</font></span>: <font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c13">b</font> = 
<font color="Maroon" title="c12">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" onmouseover="rs('rlvect_1/T16')" onmouseout="rh()">RLVECT_1:16</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">T</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c14">a1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">b1</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c14">a1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c15">b1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" onmouseover="rs('rlvect_1/T16')" onmouseout="rh()">RLVECT_1:16</a></span>
;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="normsp_2.html#K1" title="NORMSP_2:func.1">distance_by_norm_of</a> <font color="Maroon" title="c10">RNS</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font>) = 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c13">b</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D1" onmouseover="rs('normsp_2/D1')" onmouseout="rh()">NORMSP_2:def 1</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">a1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c15">b1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E6:17_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="normsp_2.html#K1" title="NORMSP_2:func.1">distance_by_norm_of</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D1" onmouseover="rs('normsp_2/D1')" onmouseout="rh()">NORMSP_2:def 1</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/>


</div><span class="kw">end;</span></div>
<a NAME="E10:17_1"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
 <a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c10">RNS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c3">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E6:17_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="binop_1.html#T2" onmouseover="rs('binop_1/T2')" onmouseout="rh()">BINOP_1:2</a></span>;<br/>
<a NAME="E11:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A13</font></span>: 
 <a href="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Maroon" title="c11">Fx0</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="c9">Fx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="ascoli.html#T1" onmouseover="rs('ascoli/T1')" onmouseout="rh()">ASCOLI:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">ClHx</font> =  <a href="ascoli.html#K1" title="ASCOLI:func.1">Cl</a> <font color="Maroon" title="c7">Hx</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> <font color="Maroon" title="c10">RNS</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">K</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E14:17_1"/><span class="lab"><font color="Green" title="E26">A14</font></span>: 
 for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  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>  st <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">G</font> holds <br/> for <font color="Olive" title="b2">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds  <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">x0</font> = <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E27">A15</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1_2"><span class="kw">now </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">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">RNS</font>  st <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">Fx0</font> holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">K</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">RNS</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"> ( <font color="Maroon" title="c15">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">Fx0</font> implies <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c15">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">K</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_1_2"/>
<font color="Maroon" title="c15">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">Fx0</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"> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c15">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">K</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font><span class="kw"> such that </span><br/><a NAME="E3:17_1_2"/><span class="lab"><font color="Green" title="E28">A16</font></span>: 
( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c16">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">g</font> = <font color="Maroon" title="c16">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a></span>;<br/><a NAME="E5:17_1_2"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x0</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c15">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_2"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="txt" href="#E6:17_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:17_1_2"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c15">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E14:17_1"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E3:17_1_2"><span class="lab"><font color="Green" title="E28">A16</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/></div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c15">r</font> = <font color="Maroon" title="c13">K</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1;<br/>
<a NAME="E17:17_1"/>
 for <font color="Olive" title="b1">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">RNS</font>  st <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Maroon" title="c11">Fx0</font> holds <br/><span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">K</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">RNS</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"> ( <font color="Maroon" title="c16">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Maroon" title="c11">Fx0</font> implies <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c16">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">K</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_3"/>
<font color="Maroon" title="c16">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Maroon" title="c11">Fx0</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"> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c16">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">K</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">seq</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c10">RNS</font><span class="kw"> such that </span><br/><a NAME="E3:17_1_3"/><span class="lab"><font color="Green" title="E28">A17</font></span>: 
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Fx0</font> &amp; <font color="Maroon" title="c17">seq</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <font color="Maroon" title="c17">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_3.html#T6" onmouseover="rs('normsp_3/T6')" onmouseout="rh()">NORMSP_3:6</a></span>;<br/>
<a NAME="E4:17_1_3"/><span class="lab"><font color="Green" title="E29">A18</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">RNS</font> : <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c10">RNS</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">K</font> </span> } </span>  is   <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</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#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c10">RNS</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#T9" onmouseover="rs('normsp_2/T9')" onmouseout="rh()">NORMSP_2:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">B</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">RNS</font> : <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c10">RNS</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">K</font> </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c10">RNS</font> ;<br/>
<a NAME="E6:17_1_3"/><span class="lab"><font color="Green" title="E30">A19</font></span>: 
<font color="Maroon" title="c18">B</font> is  <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#T15" onmouseover="rs('normsp_2/T15')" onmouseout="rh()">NORMSP_2:15</a>, <a class="txt" href="#E4:17_1_3"><span class="lab"><font color="Green" title="E29">A18</font></span></a></span>;<br/>
<a NAME="E7:17_1_3"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">B</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">seq</font> or <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_3_1"/><span class="lab"><font color="Green" title="E31">A20</font></span>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">seq</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"> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">z1</font> = <font color="Maroon" title="c19">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">RNS</font> ;<br/>
<a NAME="E3:17_1_3_1"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c20">z1</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:17_1"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E3:17_1_3"><span class="lab"><font color="Green" title="E28">A17</font></span></a>, <a class="txt" href="#E1:17_1_3_1"><span class="lab"><font color="Green" title="E31">A20</font></span></a></span>;<br/>
<a NAME="E4:17_1_3_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c10">RNS</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c20">z1</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#T2" onmouseover="rs('normsp_1/T2')" onmouseout="rh()">NORMSP_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:17_1_3_1"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:17_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c18">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_3"><span class="lab"><font color="Green" title="E28">A17</font></span></a>, <a class="txt" href="#E6:17_1_3"><span class="lab"><font color="Green" title="E30">A19</font></span></a></span>;<br/>
<a NAME="E9:17_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">RNS</font> st <br/>( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z</font> &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c10">RNS</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">z</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">K</font> )
 
;<br/>
<a NAME="E10:17_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A21</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c16">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#T2" onmouseover="rs('normsp_1/T2')" onmouseout="rh()">NORMSP_1:2</a></span>;<br/>
<a NAME="E11:17_1_3"/>
<font color="Maroon" title="c13">K</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:17_1_3"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c16">y</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">K</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:17_1_3"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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/>


</div><span class="kw">end;</span></div>
<a NAME="E18:17_1"/><span class="kw">then </span>
 <a href="normsp_3.html#K1" title="NORMSP_3:func.1">Cl</a> <font color="Maroon" title="c11">Fx0</font> is  <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="real_ns3.html#T38" onmouseover="rs('real_ns3/T38')" onmouseout="rh()">REAL_NS3:38</a></span>;<br/>
<a NAME="E19:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">ClHx</font> is  <a href="topmetr4.html#V1" title="TOPMETR4:attr.1">sequentially_compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E11:17_1"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="topmetr4.html#T18" onmouseover="rs('topmetr4/T18')" onmouseout="rh()">TOPMETR4:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:17_1"/>
<span class="p1">(<span class="default"><a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <span class="p1">(<span class="default"><a href="ascoli.html#K1" title="ASCOLI:func.1">Cl</a> <font color="Maroon" title="c7">Hx</font></span>)</span> is  <a href="topmetr.html#V2" title="TOPMETR:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:17_1"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="topmetr4.html#T14" onmouseover="rs('topmetr4/T14')" onmouseout="rh()">TOPMETR4:14</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:17"/>
<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> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c5">H</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="ascoli.html#T16" onmouseover="rs('ascoli/T16')" onmouseout="rh()">ASCOLI:16</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/>


</div>
