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<b>let </b><font color="Maroon" title="c1">CNS</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">RNS</font> being  <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/> for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Olive" title="b1">RNS</font>,<font color="Maroon" title="c1">CNS</font>  st  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">f</font> is <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a> &amp; <font color="Olive" title="b2">f</font> <a href="ncfcont1.html#R9" title="NCFCONT1:pred.9">is_continuous_on</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">f</font> holds <br/> ex <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">RNS</font> st <br/>( <font color="Olive" title="b3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">f</font> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">RNS</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">RNS</font>,<font color="Maroon" title="c1">CNS</font>  st  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font> is <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a> &amp; <font color="Olive" title="b1">f</font> <a href="ncfcont1.html#R9" title="NCFCONT1:pred.9">is_continuous_on</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font> holds <br/> ex <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">RNS</font> st <br/>( <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">RNS</font>,<font color="Maroon" title="c1">CNS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> is <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a> &amp; <font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#R9" title="NCFCONT1:pred.9">is_continuous_on</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> implies  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">RNS</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> ) )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:134"/><i><font color="Green" title="E101">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:134"/><i><font color="Green" title="E102">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> is <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a>
 <b>and </b><br/><a NAME="E3:134"/><i><font color="Green" title="E103">A3</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#R9" title="NCFCONT1:pred.9">is_continuous_on</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">RNS</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> )</span><br/>

<a NAME="E4:134"/><i><font color="Green" title="E104">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> is <a href="nfcont_1.html#V1" title="NFCONT_1:attr.1">compact</a>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E2:134"><i><font color="Green" title="E102">A2</font></i></a>, <a class="ref" href="ncfcont1.html#D4" target="_self" title="NCFCONT1:def.4">Def4</a></i>;<br/>
<a NAME="E5:134"/><i><font color="Green" title="E105">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span>
 
<b>by </b><i><a class="ref" href="ncfcont1.html#D4" target="_self" title="NCFCONT1:def.4">Def4</a></i>;<br/>
<a NAME="E6:134"/><b>then </b><i><font color="Green" title="E106">A6</font></i>: 
<span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="nfcont_1.html#R4" title="NFCONT_1:pred.4">is_continuous_on</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E3:134"><i><font color="Green" title="E103">A3</font></i></a>, <a class="ref" href="ncfcont1.html#T94" target="_self" title="NCFCONT1:th.94">Th94</a></i>;<br/>
<a NAME="E7:134"/><i><font color="Green" title="E107">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E1:134"><i><font color="Green" title="E101">A1</font></i></a>, <a class="txt" href="ncfcont1.html#E5:134"><i><font color="Green" title="E105">A5</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<a NAME="E8:134"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E4:134"><i><font color="Green" title="E104">A4</font></i></a>, <a class="txt" href="ncfcont1.html#E6:134"><i><font color="Green" title="E106">A6</font></i></a>, <a class="ref" href="nfcont_1.html#T38" title="NFCONT_1:th.38">NFCONT_1:38</a></i>;<br/>
<a NAME="E9:134"/><b>then </b><i><font color="Green" title="E108">A8</font></i>: 
(  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> )
 
<b>by </b><i><a class="txt" href="ncfcont1.html#E7:134"><i><font color="Green" title="E107">A7</font></i></a>, <a class="ref" href="rcomp_1.html#T32" title="RCOMP_1:th.32">RCOMP_1:32</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">RNS</font><b> such that </b><br/><a NAME="E11:134"/><i><font color="Green" title="E109">A9</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">RNS</font><b> such that </b><br/><a NAME="E13:134"/><i><font color="Green" title="E110">A10</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E9:134"><i><font color="Green" title="E108">A8</font></i></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">RNS</font> st <br/>( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b1">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> )</span><br/>

<b>take </b>
<font color="Maroon" title="c5">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> )</span><br/>

<b>thus </b><a NAME="E14:134"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K3" title="NCFCONT1:func.3">.||</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E11:134"><i><font color="Green" title="E109">A9</font></i></a>, <a class="txt" href="ncfcont1.html#E13:134"><i><font color="Green" title="E110">A10</font></i></a>, <a class="ref" href="ncfcont1.html#D4" target="_self" title="NCFCONT1:def.4">Def4</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
