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

<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">L</font> be   <a href="hahnban.html#NM1">Functional</a> of <font color="Maroon">X</font>;<br/>





<b>thus </b><a NAME="E1:4"/>
( <font color="Maroon">Y</font> <a href="bhsp_6.html#R1">is_summable_set_by</a> <font color="Maroon">L</font> implies for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> holds <br/> ex <font color="Olive">Y0</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon">Y</font> <a href="bhsp_6.html#R1">is_summable_set_by</a> <font color="Maroon">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:4_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">Y0</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y1</font> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> ) )
 <b>by </b><i><a class="ref" href="bhsp_6.html#D6">BHSP_6:def 6</a></i>;<br/>
<a NAME="E4:4_1"/>
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> holds <br/> ex <font color="Olive">Y0</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:4_1_1"/><i><font color="Green">E39</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/>

<a NAME="E2:4_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y0</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:4_1_1"/><i><font color="Green">E53</font></i>: 
( not <font color="Maroon">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y1</font> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2 ) )
 <b>by </b><i/>;<br/>
<a NAME="E5:4_1_1"/>
for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y1</font> be  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:4_1_1_1"/><i><font color="Green">E54</font></i>: 
( not <font color="Maroon">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y1</font> )
 ;<br/>

<b>set </b><font color="Maroon">Y1'</font> = <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y1</font>;<br/>
<a NAME="E2:4_1_1_1"/>
( <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y1</font> &amp; <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y1</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y1</font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E3:4_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <span class="p3">(<span class="default"><font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i/>;<br/>
<a NAME="E4:4_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <span class="p3">(<span class="default"><font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/>
<a NAME="E5:4_1_1_1"/><i><font color="Green">E57</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i/>;<br/>
<a NAME="E6:4_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b> <a href="bhsp_5.html#K5">setopfunc</a> <span class="p1">(<span class="default"><font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a>  = 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="bhsp_5.html#T14">BHSP_5:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
;<br/>
<a NAME="E8:4_1_1_1"/><b>then </b>
 <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <span class="p2">(<span class="default"><font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
 
;<br/>
<b>hence </b><a NAME="E9:4_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:4_1_1"/>
 ex <font color="Olive">Y0</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:4_1"/>
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> holds <br/> ex <font color="Olive">Y0</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> ) )
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:4"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> holds <br/> ex <font color="Olive">Y0</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> ) )
 ;<br/>

<a NAME="E3:4"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> holds <br/> ex <font color="Olive">Y0</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y1</font> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> &amp; not <font color="Maroon">a<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">z</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="real_1.html#K3">+</a> 1</span>)</span> ) );<br/>
<a NAME="E1:4_2"/><i><font color="Green">E60</font></i>: 
 ex <font color="Olive">B</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font> st <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_2_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">e</font> = 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E4:4_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xx</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E5:4_2_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">xx</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:4_2_1_1"/><b>then </b>
0 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>then consider </b><font color="Maroon">Y0</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:4_2_1_1"/><i><font color="Green">E64</font></i>: 
( not <font color="Maroon">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> ) )
 <b>by </b><i/>;<br/><a NAME="E9:4_2_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">Y0</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/><a NAME="E10:4_2_1_1"/>
for <font color="Olive">z</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> holds <br/>for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E11:4_2_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font> &amp; <font color="Olive">y</font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> &amp; not <font color="Olive">y</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">z</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> holds <br/>for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">y</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="real_1.html#K3">+</a> 1</span>)</span> ) )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E2:4_2_1"/><b>then </b><i><font color="Green">E66</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
;<br/>
<b>thus </b><a NAME="E3:4_2_1"/>
 ex <font color="Olive">B</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font> st <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();</i><br/>


</div><b>end;</b></div>
<a NAME="E2:4_2"/>
 ex <font color="Olive">A</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font> st <br/>for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> &amp; not <font color="Olive">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> holds <br/><font color="Olive">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">B</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E2:4_2_2"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/>( <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> &amp; not <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">z</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> holds <br/>for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="real_1.html#K3">+</a> 1</span>)</span> ) )
 <b>by </b><i/>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E3:4_2_2"/>
 ex <font color="Olive">A</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Olive">A</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>,<font color="Olive">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>) ) )
 
<b>from </b><i><a class="ref" href="recdef_1.html#S3">RECDEF_1:sch 3</a>();<br/></i>
<b>then consider </b><font color="Maroon">A</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:4_2_2"/><i><font color="Green">E72</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> ) )
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> &amp; not <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> ) );<br/>
<a NAME="E6:4_2_2"/>
for <font color="Olive">j0</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j0</font> <a href="hidden.html#R1">=</a> 0 holds <br/>for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j0</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:4_2_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">j0</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">j0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E2:4_2_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:4_2_2_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">j</font>]
 ;<br/><a NAME="E2:4_2_2_1_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:4_2_2_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:4_2_2_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T18">NAT_1:18</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E4:4_2_2_1"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div><b>end;</b></div>
<a NAME="E7:4_2_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:4_2_2_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>]
 ;<br/><a NAME="E2:4_2_2_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:4_2_2_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><a NAME="E4:4_2_2_2"/><i><font color="Green">E84</font></i>: 
 ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:4_2_2_2"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y1</font> be  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:4_2_2_2_1"/><i><font color="Green">E86</font></i>: 
( not <font color="Maroon">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y1</font> )
 ;<br/>

<a NAME="E2:4_2_2_2_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:4_2_2_2_1"/><b>then </b>
<font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:4_2_2_2_1"/><b>then </b>
( not <font color="Maroon">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y1</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<b>hence </b><a NAME="E5:4_2_2_2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E6:4_2_2_2"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_2_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:4_2_2_2_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E2:4_2_2_2_2"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_2_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:4_2_2_2_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">j</font>]
 <b>and </b><br/><a NAME="E2:4_2_2_2_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_2_2_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_2_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_2_2_2_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_2_2_2_1_1_1_1"><b>case </b></a><a NAME="E1:4_2_2_2_2_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_2_2_2_2_1_1_1_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_2_2_2_1_1_1_2"><b>case </b></a><a NAME="E1:4_2_2_2_2_1_1_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:4_2_2_2_2_1_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:4_2_2_2_2_1_1_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:4_2_2_2_2_1_1_1_2"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:4_2_2_2_2_1_1_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E6:4_2_2_2_2_1_1_1_2"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:4_2_2_2_2_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:4_2_2_2_2"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:4_2_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , , , , <a class="ref" href="finset_1.html#T14">FINSET_1:14</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:4_2_2"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E10:4_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_2_2_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_2_2_3"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E5:4_2_2_3"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:4_2_2_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:4_2_2"/><b>then </b>
<font color="Maroon">A</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>hence </b><a NAME="E12:4_2_2"/>
 ex <font color="Olive">A</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font> st <br/>for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> &amp; not <font color="Olive">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> holds <br/><font color="Olive">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">A</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:4_2"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> &amp; not <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> ) )
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">Y1</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a>  );<br/>
<a NAME="E5:4_2"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:4_2_3"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E3:4_2_3"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> &amp; not <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> ) )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">Y1</font> = <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">y</font> =  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a>  as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E6:4_2_3"/>
 ex <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:4_2_3"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:4_2"/>
 ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a>  st <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E8:4_2"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/> ex <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y1</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a>  )
 ;<br/>
<b>set </b><font color="Maroon">seq</font> = <font color="Maroon">F</font>;<br/>
<a NAME="E9:4_2"/><i><font color="Green">E103</font></i>: 
for <font color="Olive">e</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:4_2_4"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:4_2_4"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:4_2_4"/><i><font color="Green">E106</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E5:4_2_4"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> )
 ;<br/>

<b>consider </b><font color="Maroon">Y0</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:4_2_4"/><i><font color="Green">E108</font></i>: 
( not <font color="Maroon">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y0</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a>  )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Y1</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:4_2_4"/><i><font color="Green">E109</font></i>: 
( not <font color="Maroon">Y1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a>  )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Y2</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E11:4_2_4"/><i><font color="Green">E111</font></i>: 
( not <font color="Maroon">Y2</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E12:4_2_4"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y1</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E13:4_2_4"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font>
 
<b>by </b><i>, , , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_4_1_1"/>
( <font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> or <font color="Maroon">Y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_4_1_1_1"><b>case </b></a><a NAME="E1:4_2_4_1_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_4_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_4_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_4_1_1_1_1_1"/>
( <font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font> or <font color="Maroon">Y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_4_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:4_2_4_1_1_1_1_1_1"/>
<font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_2_4_1_1_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , , , <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_4_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:4_2_4_1_1_1_1_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">Y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y2</font>
 ;<br/><div class="add"><a NAME="E2:4_2_4_1_1_1_1_1_2"/>
 ex <font color="Olive">Y02</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y02</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y02</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">Y02</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y0</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">Y02</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_4_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">Y02</font> = <font color="Maroon">Y2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Y0</font>;<br/>

<a NAME="E1:4_2_4_1_1_1_1_1_2_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">Y2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y02</font> = 
<font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">Y2</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<b>hence </b><a NAME="E3:4_2_4_1_1_1_1_1_2_1"/>
( not <font color="Maroon">Y02</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y02</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y02</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y0</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y02</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font> )
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">Y02</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:4_2_4_1_1_1_1_1_2"/><i><font color="Green">E118</font></i>: 
( not <font color="Maroon">Y02</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y02</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y02</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y0</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y02</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font> )
 ;<br/><a NAME="E5:4_2_4_1_1_1_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><i><font color="Green">E119</font></i>:  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a>  = 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y02</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="bhsp_5.html#T14">BHSP_5:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y02</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
;<br/><a NAME="E7:4_2_4_1_1_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y02</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E8:4_2_4_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y02</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><i><font color="Green">E121</font></i>:  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y02</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y02</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
;<br/><a NAME="E10:4_2_4_1_1_1_1_1_2"/>
<font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/><b>hence </b><a NAME="E11:4_2_4_1_1_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:4_2_4_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_4_1_1_2"><b>case </b></a><a NAME="E1:4_2_4_1_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">Y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_4_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_4_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_4_1_1_2_1_1"/>
( <font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font> or <font color="Maroon">Y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_4_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:4_2_4_1_1_2_1_1_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font>
 ;<br/><div class="add"><a NAME="E2:4_2_4_1_1_2_1_1_1"/>
 ex <font color="Olive">Y01</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y01</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y01</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">Y01</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y0</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">Y01</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_4_1_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">Y01</font> = <font color="Maroon">Y1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Y0</font>;<br/>

<a NAME="E1:4_2_4_1_1_2_1_1_1_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">Y1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y01</font> = 
<font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y1</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">Y1</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<b>hence </b><a NAME="E3:4_2_4_1_1_2_1_1_1_1"/>
( not <font color="Maroon">Y01</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y01</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y01</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y0</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y01</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> )
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">Y01</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:4_2_4_1_1_2_1_1_1"/><i><font color="Green">E126</font></i>: 
( not <font color="Maroon">Y01</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y01</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y01</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y0</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y01</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> )
 ;<br/><a NAME="E5:4_2_4_1_1_2_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><i><font color="Green">E127</font></i>:  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a>  = 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y01</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="bhsp_5.html#T14">BHSP_5:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y01</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
;<br/><a NAME="E7:4_2_4_1_1_2_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y01</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E8:4_2_4_1_1_2_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i>, , , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:4_2_4_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E10:4_2_4_1_1_2_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_4_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:4_2_4_1_1_2_1_1_2"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">Y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y2</font>
 ;<br/><div class="add"><a NAME="E2:4_2_4_1_1_2_1_1_2"/>
 ex <font color="Olive">Y01</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y01</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y01</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">Y01</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y0</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">Y01</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_4_1_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">Y01</font> = <font color="Maroon">Y1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Y0</font>;<br/>

<a NAME="E1:4_2_4_1_1_2_1_1_2_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">Y1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y01</font> = 
<font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y1</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">Y1</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<b>hence </b><a NAME="E3:4_2_4_1_1_2_1_1_2_1"/>
( not <font color="Maroon">Y01</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y01</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y01</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y0</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y01</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> )
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">Y01</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:4_2_4_1_1_2_1_1_2"/><i><font color="Green">E130</font></i>: 
( not <font color="Maroon">Y01</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y01</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y01</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y0</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y01</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> )
 ;<br/><a NAME="E5:4_2_4_1_1_2_1_1_2"/>
 ex <font color="Olive">Y02</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y02</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y02</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">Y02</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y0</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">Y02</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_4_1_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">Y02</font> = <font color="Maroon">Y2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Y0</font>;<br/>

<a NAME="E1:4_2_4_1_1_2_1_1_2_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">Y2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y02</font> = 
<font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">Y2</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<b>hence </b><a NAME="E3:4_2_4_1_1_2_1_1_2_2"/>
( not <font color="Maroon">Y02</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y02</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y02</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y0</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y02</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font> )
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">Y02</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:4_2_4_1_1_2_1_1_2"/><i><font color="Green">E132</font></i>: 
( not <font color="Maroon">Y02</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y02</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y02</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y0</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y02</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y2</font> )
 ;<br/><a NAME="E8:4_2_4_1_1_2_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y01</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E9:4_2_4_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E133</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y01</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:4_2_4_1_1_2_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y02</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E11:4_2_4_1_1_2_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y02</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:4_2_4_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y01</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y02</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E13:4_2_4_1_1_2_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><i><font color="Green">E135</font></i>:  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a>  = 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y01</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="bhsp_5.html#T14">BHSP_5:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y01</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
;<br/><a NAME="E15:4_2_4_1_1_2_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><i><font color="Green">E136</font></i>:  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a>  = 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>,<span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y02</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="bhsp_5.html#T14">BHSP_5:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y02</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
;<br/><a NAME="E17:4_2_4_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y01</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y02</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>
 <b>by </b><i>, , , </i>;<br/><a NAME="E18:4_2_4_1_1_2_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y01</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y02</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T143">COMPLEX1:143</a></i>;<br/><b>hence </b><a NAME="E19:4_2_4_1_1_2_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:4_2_4_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E15:4_2_4"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:4_2"/>
for <font color="Olive">e</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> holds <br/> ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:4_2_5"/><i><font color="Green">E137</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/>

<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:4_2_5"/><i><font color="Green">E138</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E4:4_2_5"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:4_2_5"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:4_2"/><b>then </b>
<font color="Maroon">F</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T58">SEQ_4:58</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E13:4_2"/><i><font color="Green">E139</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font>
 <b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E15:4_2"/><i><font color="Green">E140</font></i>: 
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> holds <br/> ex <font color="Olive">Y0</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y1</font> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:4_2_6"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:4_2_6"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:4_2_6"/><i><font color="Green">E143</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:4_2_6"/><i><font color="Green">E144</font></i>: 
( 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">Y0</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:4_2_6"/><i><font color="Green">E145</font></i>: 
( not <font color="Maroon">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y0</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E9:4_2_6"/><i><font color="Green">E146</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y1</font> be  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:4_2_6_1"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y1</font> &amp; <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_6_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_6_1_1_1"/>
( <font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> or <font color="Maroon">Y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_6_1_1_1_1"><b>case </b></a><a NAME="E1:4_2_6_1_1_1_1"/>
<font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font>
 ;<br/><div class="add"><a NAME="E2:4_2_6_1_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, <a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/><a NAME="E3:4_2_6_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E4:4_2_6_1_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_6_1_1_1_2"><b>case </b></a><a NAME="E1:4_2_6_1_1_1_2"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">Y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y1</font>
 ;<br/><div class="add"><a NAME="E2:4_2_6_1_1_1_2"/>
 ex <font color="Olive">Y2</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y2</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y2</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">Y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_6_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">Y2</font> = <font color="Maroon">Y1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Y0</font>;<br/>

<a NAME="E1:4_2_6_1_1_1_2_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">Y1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font> = 
<font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y1</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">Y1</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<b>hence </b><a NAME="E3:4_2_6_1_1_1_2_1"/>
( not <font color="Maroon">Y2</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y2</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> )
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">Y2</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:4_2_6_1_1_1_2"/><i><font color="Green">E150</font></i>: 
( not <font color="Maroon">Y2</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y0</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y2</font> &amp; <font color="Maroon">Y0</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> )
 ;<br/><a NAME="E5:4_2_6_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:4_2_6_1_1_1_2"/><b>then </b><i><font color="Green">E151</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:4_2_6_1_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>,<span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="bhsp_5.html#T14">BHSP_5:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span>

;<br/><a NAME="E9:4_2_6_1_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T21">ABSVALUE:21</a></i>;<br/><a NAME="E10:4_2_6_1_1_1_2"/><b>then </b><i><font color="Green">E152</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/><a NAME="E11:4_2_6_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y0</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E12:4_2_6_1_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:4_2_6_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:4_2_6"/>
 ex <font color="Olive">Y0</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y1</font> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<b>thus </b><a NAME="E16:4_2"/>
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> holds <br/> ex <font color="Olive">Y0</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( not <font color="Olive">Y0</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">Y1</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">Y0</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y1</font> &amp; <font color="Olive">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="bhsp_5.html#K5">setopfunc</a> <font color="Olive">Y1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">L</font>,<a href="binop_2.html#K33">addreal</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:4"/>
<font color="Maroon">Y</font> <a href="bhsp_6.html#R1">is_summable_set_by</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="bhsp_6.html#D6">BHSP_6:def 6</a></i>;<br/>


</div>
