<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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">Y</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="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<font color="Olive" title="b1">Y</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> holds <br/> <a href="ordeq_01.html#K4" title="ORDEQ_01:func.4">modetrans</a> (<font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">Y</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</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="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<font color="Maroon" title="c2">Y</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> holds <br/> <a href="ordeq_01.html#K4" title="ORDEQ_01:func.4">modetrans</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="nfcont_3.html#V1" title="NFCONT_3:attr.1">continuous</a>  <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<font color="Maroon" title="c2">Y</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="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> implies  <a href="ordeq_01.html#K4" title="ORDEQ_01:func.4">modetrans</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:25"/>
 <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> <font color="Maroon" title="c1">X</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="ordeq_01.html#K4" title="ORDEQ_01:func.4">modetrans</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f</font></span><br/>

<a NAME="E2:25"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordeq_01.html#K1" title="ORDEQ_01:func.1">ContinuousFunctions</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordeq_01.html#D2" target="_self" onmouseover="rs('ordeq_01/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:25"/>
 <a href="ordeq_01.html#K4" title="ORDEQ_01:func.4">modetrans</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordeq_01.html#D5" target="_self" onmouseover="rs('ordeq_01/D5')" onmouseout="rh()">Def5</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>
