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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>  &amp; <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  holds <br/> <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">f</font> is  <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">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"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  implies  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> is  <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:33"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> 
 <span class="kw">and </span><br/><a NAME="E2:33"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> is  <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> is  <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1"><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">s1</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">S</font>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> holds <br/> ex <font color="Olive" title="b2">q2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive" title="b2">q2</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Olive" title="b1">s1</font> &amp; <font color="Olive" title="b2">q2</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="Olive" title="b2">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">s1</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">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"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> implies  ex <font color="Olive" title="b1">q2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive" title="b1">q2</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c3">s1</font> &amp; <font color="Olive" title="b1">q2</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="Olive" title="b1">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:33_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">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">  ex <font color="Olive" title="b1">q2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive" title="b1">q2</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c3">s1</font> &amp; <font color="Olive" title="b1">q2</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="Olive" title="b1">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> )</span><br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>( $2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/><a NAME="E2:33_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">p</font>]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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">  ex <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">p</font>]</span><br/>

<a NAME="E1:33_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:33_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">s1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E4:33_1_1"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
( <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c3">s1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="partfun1.html#T3" onmouseover="rs('partfun1/T3')" onmouseout="rh()">PARTFUN1:3</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">p</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">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">p</font>]</span><br/>

<span class="kw">thus </span><a NAME="E5:33_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">p</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1_1"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</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">consider </span><font color="Maroon" title="c4">q1</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E4:33_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">q1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:33_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>);<br/></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</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="c4">q1</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c5">x</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="c4">q1</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> )</span><br/><span class="kw">assume </span><a NAME="E1:33_1_2"/>
<font color="Maroon" title="c5">x</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="c4">q1</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span><br/><a NAME="E2:33_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" onmouseover="rs('funct_2/T113')" onmouseout="rh()">FUNCT_2:113</a></span>;<br/><span class="kw">hence </span><a NAME="E3:33_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E21">A6</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><a NAME="E6:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A7</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">s2</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><span class="kw"> such that </span><br/><a NAME="E8:33_1"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon" title="c5">s2</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c4">q1</font>
 <span class="kw">and </span><br/><a NAME="E9:33_1"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
<font color="Maroon" title="c5">s2</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E10:33_1"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c5">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="rcomp_1.html#D3" onmouseover="rs('rcomp_1/D3')" onmouseout="rh()">RCOMP_1:def 3</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c6">q2</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">s2</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="c6">q2</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c6">q2</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="c6">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> )</span><br/><a NAME="E11:33_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">s2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="valued_0.html#T21" onmouseover="rs('valued_0/T21')" onmouseout="rh()">VALUED_0:21</a></span>;<br/><a NAME="E12:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A11</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">s2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1_3"><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">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">q1</font></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Olive" title="b1">n</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">q1</font></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c7">n</font></span><br/><a NAME="E1:33_1_3"/>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:33_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">q1</font></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="funct_2.html#T109" onmouseover="rs('funct_2/T109')" onmouseout="rh()">FUNCT_2:109</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="E14:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">q1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" onmouseover="rs('funct_2/T63')" onmouseout="rh()">FUNCT_2:63</a></span>;<br/><a NAME="E15:33_1"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c5">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:33_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a></span>;<br/><a NAME="E16:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nfcont_3.html#R1" title="NFCONT_3:pred.1">is_continuous_in</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c5">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/><a NAME="E17:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="nfcont_3.html#R1" title="NFCONT_3:pred.1">is_continuous_in</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c5">s2</font>
 ;<br/><a NAME="E18:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Maroon" title="c5">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K5" title="NORMSP_1:func.5">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c5">s2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:33_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E19:33_1"/>
( <font color="Maroon" title="c6">q2</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c3">s1</font> &amp; <font color="Maroon" title="c6">q2</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="c6">q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E14:33_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E8:33_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E9:33_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E17:33_1"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a>, <a class="ref" href="valued_0.html#T22" onmouseover="rs('valued_0/T22')" onmouseout="rh()">VALUED_0:22</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="E4:33"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">f</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="ref" href="nfcont_1.html#D2" onmouseover="rs('nfcont_1/D2')" onmouseout="rh()">NFCONT_1:def 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>
