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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<a NAME="E1:6"/>
(  ex <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> ) iff  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:6_1_1"/>
 ex <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 ;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:6_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 ;<br/><a NAME="E4:6_1_1"/>
 <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:6_1_1"/>
 ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:6_1"/>
(  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">a</font> ) implies  ex <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc6.html#K2" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font> iff ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> ) ) )
 <b>by </b><i><a class="ref" href="mesfunc1.html#D13">MESFUNC1:def 13</a></i>;<br/>


</div>
