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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="Maroon" title="c1">CNS</font>,<font color="Olive" title="b1">RNS</font><br/> for <font color="Olive" title="b3">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">CNS</font>  st <font color="Olive" title="b3">Y</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; <font color="Olive" title="b3">Y</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="Olive" title="b2">f</font> &amp; <font color="Olive" title="b3">Y</font> is  <a href="ncfcont1.html#V1" title="NCFCONT1:attr.1">compact</a>  &amp; <font color="Olive" title="b2">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b3">Y</font> holds <br/> ex <font color="Olive" title="b4">x1</font>, <font color="Olive" title="b5">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</font> st <br/>( <font color="Olive" title="b4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">Y</font> &amp; <font color="Olive" title="b5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">Y</font> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">Y</font></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">Y</font></span>)</span> )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c1">CNS</font>,<font color="Maroon" title="c2">RNS</font><br/> for <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">CNS</font>  st <font color="Olive" title="b2">Y</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; <font color="Olive" title="b2">Y</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="Olive" title="b1">f</font> &amp; <font color="Olive" title="b2">Y</font> is  <a href="ncfcont1.html#V1" title="NCFCONT1:attr.1">compact</a>  &amp; <font color="Olive" title="b1">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b2">Y</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="Maroon" title="c1">CNS</font> st <br/>( <font color="Olive" title="b3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Y</font> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">Y</font></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">Y</font></span>)</span> )</span><br/><span class="kw">let </span><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="c1">CNS</font>,<font color="Maroon" title="c2">RNS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">CNS</font>  st <font color="Olive" title="b1">Y</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; <font color="Olive" title="b1">Y</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="c3">f</font> &amp; <font color="Olive" title="b1">Y</font> is  <a href="ncfcont1.html#V1" title="NCFCONT1:attr.1">compact</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Olive" title="b1">Y</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="c1">CNS</font> st <br/>( <font color="Olive" title="b2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">Y</font></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">Y</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">CNS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">Y</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; <font color="Maroon" title="c4">Y</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="c3">f</font> &amp; <font color="Maroon" title="c4">Y</font> is  <a href="ncfcont1.html#V1" title="NCFCONT1:attr.1">compact</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c4">Y</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="c1">CNS</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:133"/><span class="lab"><font color="Green" title="E79">A1</font></span>: 
<font color="Maroon" title="c4">Y</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> 
 <span class="kw">and </span><br/><a NAME="E2:133"/><span class="lab"><font color="Green" title="E80">A2</font></span>: 
<font color="Maroon" title="c4">Y</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="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E3:133"/><span class="lab"><font color="Green" title="E81">A3</font></span>: 
<font color="Maroon" title="c4">Y</font> is  <a href="ncfcont1.html#V1" title="NCFCONT1:attr.1">compact</a> 
 <span class="kw">and </span><br/><a NAME="E4:133"/><span class="lab"><font color="Green" title="E82">A4</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c4">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c1">CNS</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span> )</span><br/>

<span class="lab"><font color="Green" title="E83">A5</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">Y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
<font color="Maroon" title="c4">Y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E6:133"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font> <a href="ncfcont1.html#R8" title="NCFCONT1:pred.8">is_continuous_on</a> <font color="Maroon" title="c4">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:133_2"/>
<font color="Maroon" title="c4">Y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:133"><span class="lab"><font color="Green" title="E83">A5</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ncfcont1.html#D22" target="_self" title="NCFCONT1:def.22">NCFCONT1:def 22</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</font>  st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font> <a href="ncfcont1.html#R2" title="NCFCONT1:pred.2">is_continuous_in</a> <font color="Olive" title="b1">x0</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">r</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font> <a href="ncfcont1.html#R2" title="NCFCONT1:pred.2">is_continuous_in</a> <font color="Maroon" title="c5">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:133_2"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font> <a href="ncfcont1.html#R2" title="NCFCONT1:pred.2">is_continuous_in</a> <font color="Maroon" title="c5">r</font></span><br/>

<a NAME="E3:133_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font> <a href="ncfcont1.html#R2" title="NCFCONT1:pred.2">is_continuous_in</a> <font color="Maroon" title="c5">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:133"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="ref" href="ncfcont1.html#D22" target="_self" title="NCFCONT1:def.22">Def22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:133_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font> <a href="ncfcont1.html#R2" title="NCFCONT1:pred.2">is_continuous_in</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T101" title="RELAT_1:th.101">RELAT_1:101</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">CNS</font><span class="kw"> such that </span><br/><a NAME="E8:133"/><span class="lab"><font color="Green" title="E84">A6</font></span>: 
<font color="Maroon" title="c5">x1</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="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:133"/><span class="lab"><font color="Green" title="E85">A7</font></span>: 
<font color="Maroon" title="c6">x2</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="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:133"/><span class="lab"><font color="Green" title="E86">A8</font></span>: 
( <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">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#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E3:133"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E5:133"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="ref" href="ncfcont1.html#T109" target="_self" title="NCFCONT1:th.109">Th109</a></span>;<br/>
<a NAME="E11:133"/><span class="lab"><font color="Green" title="E87">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c1">CNS</font> st <br/>( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Olive" title="b1">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E12:133"/>
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:133"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E8:133"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="txt" href="#E9:133"><span class="lab"><font color="Green" title="E85">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span> )</span><br/>

<span class="lab"><font color="Green" title="E88">A10</font></span>: <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> = 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></span>
<br/>.= 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ncfcont1.html#T112" target="_self" title="NCFCONT1:th.112">Th112</a></span>
;<br/>
<a NAME="E14:133"/><span class="lab"><font color="Green" title="E89">A11</font></span>: 
<font color="Maroon" title="c6">x2</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"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:133"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E90">A12</font></span>: <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x2</font> = 
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:133"><span class="lab"><font color="Green" title="E89">A11</font></span></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x2</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:133"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></span>
<br/>.= 
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E5:133"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E9:133"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="txt" href="#E11:133"><span class="lab"><font color="Green" title="E87">A9</font></span></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></span>
<br/>.= 
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E5:133"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E9:133"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="txt" href="#E11:133"><span class="lab"><font color="Green" title="E87">A9</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
;<br/>
<a NAME="E16:133"/><span class="lab"><font color="Green" title="E91">A13</font></span>: 
<font color="Maroon" title="c5">x1</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"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:133"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></span>;<br/>
<span class="kw">then </span><span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x1</font> = 
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x1</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:133"><span class="lab"><font color="Green" title="E91">A13</font></span></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x1</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:133"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></span>
<br/>.= 
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E5:133"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E8:133"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="txt" href="#E11:133"><span class="lab"><font color="Green" title="E87">A9</font></span></a>, <a class="ref" href="ncfcont1.html#D3" target="_self" title="NCFCONT1:def.3">Def3</a></span>
<br/>.= 
<span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E5:133"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E8:133"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="txt" href="#E11:133"><span class="lab"><font color="Green" title="E87">A9</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E18:133"/>
( <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span> &amp; <span class="p1"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">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"><span class="p2"><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="ncfcont1.html#K2" title="NCFCONT1:func.2">.||</a></span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:133"><span class="lab"><font color="Green" title="E86">A8</font></span></a>, <a class="txt" href="#E15:133"><span class="lab"><font color="Green" title="E90">A12</font></span></a>, <a class="txt" href="#E13:133"><span class="lab"><font color="Green" title="E88">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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