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

<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="rlsub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>

<b>let </b><font color="Maroon">q</font> be   <a href="hahnban.html#NM2">Banach-Functional</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">fi</font> be   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E2:37"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 ;<br/>

<a NAME="E3:37"/><i><font color="Green">E25</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="rlsub_2.html#K1">+</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:37"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="rlvect_3.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_4.html#T12">RLVECT_4:12</a></i>;<br/>
<a NAME="E5:37"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x0</font> = <font color="Maroon">v</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="hahnban.html#E11:37"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rlsub_2.html#T9">RLSUB_2:9</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">x1</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rlsub_1.html#T18">RLSUB_1:18</a></i>;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">y</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> </span>}</span> ;<br/>
<a NAME="E9:37"/><i><font color="Green">E36</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="supinf_1.html#K3">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:37_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> </span>}</span> 
 ;<br/>

<a NAME="E2:37_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> ex <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> )
 
;<br/>
<b>hence </b><a NAME="E3:37_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:37"/><i><font color="Green">E38</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">y</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="supinf_1.html#K3">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:37_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">y</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> </span>}</span> 
 ;<br/>

<a NAME="E2:37_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> ex <font color="Olive">y</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">y</font></span>)</span></span>)</span> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> )
 
;<br/>
<b>hence </b><a NAME="E3:37_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Maroon">x2</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">y1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">y2</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:37_3"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 ;<br/><a NAME="E2:37_3"/><b>then </b>
<font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T24">RLSUB_1:24</a></i>;<br/><a NAME="E3:37_3"/><b>then </b>
<font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y2</font></span>)</span>
 <b>by </b><i><a class="txt" href="hahnban.html#E10:37"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E4:37_3"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y2</font></span>)</span>
 <b>by </b><i/>;<br/><font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">y2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">y2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">y2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/><a NAME="E6:37_3"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hahnban.html#E3:37"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E7:37_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:37_3"/><b>then </b>
<font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E9:37_3"/><b>then </b>
<font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E10:37_3"/>
<span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">b</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>assume </b><a NAME="E1:37_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Maroon">y1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:37_4"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> )
 ;<br/><b>assume </b><a NAME="E4:37_4"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">y</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Maroon">y2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:37_4"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 ;<br/><a NAME="E7:37_4"/>
<span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hahnban.html#E3:37_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="hahnban.html#T8" target="_self">Th8</a>, </i>;<br/><b>hence </b><a NAME="E8:37_4"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T8" target="_self">Th8</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E13:37"/>
( <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> </span>}</span>  &amp; <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">y</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> </span>}</span>  )
 
;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Olive">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> </span>}</span> , <font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">y</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font>, <font color="Olive">y</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i>, <a class="txt" href="hahnban.html#E12:37"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:37"/><i><font color="Green">E112</font></i>: 
 <a href="supinf_1.html#K6">sup</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:37"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>;<br/>
<b>assume </b><a NAME="E2:37_5"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E3:37_5"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:37_5"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:37"/>
<font color="Maroon">A</font> is <a href="supinf_1.html#V3">bounded_above</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  ;<br/>
<a NAME="E3:37_6"/>
<font color="Maroon">b</font> is    <a href="supinf_1.html#M1">majorant</a> of <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a><br/>

<b>thus </b><a NAME="E1:37_6_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="supinf_1.html#M1">majorant</a> of <font color="Maroon">A</font> ;<br/>
<b>take </b>
<font color="Maroon">b</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="supinf_1.html#D11">SUPINF_1:def 11</a><br/>

<b>thus </b><a NAME="E5:37_6"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i><a class="txt" href="hahnban.html#E12:37"><i><font color="Green">E41</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">r</font> =  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">A</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="supinf_1.html#T83">SUPINF_1:83</a></i>;<br/>
<a NAME="E19:37"/>
not <font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">psi</font> being   <a href="hahnban.html#NM3">linear-Functional</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E21:37"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">psi</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font>
 <b>and </b><br/><a NAME="E22:37"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">psi</font>
;<br/>

<b>thus </b><a NAME="E23:37"/>
<font color="Maroon">psi</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T9" target="_self">Th9</a></i>;<br/>

<b>let </b><font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><a NAME="E24:37"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 ;<br/>

<a NAME="E25:37"/>
<font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X</font> <a href="rlsub_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">y2'</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E27:37"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">y1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y2'</font> <a href="rlvect_1.html#R1">in</a>  <a href="rlvect_3.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>and </b><br/><a NAME="E28:37"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y2'</font>
 <b>by </b><i><a class="ref" href="rlsub_2.html#T5">RLSUB_2:5</a></i>;<br/>
<b>consider </b><font color="Maroon">t</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E30:37"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">y2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="rlvect_4.html#T11">RLVECT_4:11</a></i>;<br/>
<a NAME="E31:37"/>
<font color="Maroon">y1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X</font> <a href="rlsub_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rlsub_2.html#T6">RLSUB_2:6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">y1</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_7"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_7"/>
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_7_1"><b>suppose </b></a><a NAME="E1:37_7_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:37_7_1"/><b>then </b>
<font color="Maroon">y2'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>;<br/><a NAME="E3:37_7_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T13" target="_self">Th13</a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/><a NAME="E4:37_7_1"/><b>then </b>
<font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="hahnban.html#E10:37"><i><font color="Green">E38</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:37_7_1"/>
<font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T9" target="_self">Th9</a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_7_2"><b>suppose </b></a><a NAME="E1:37_7_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:37_7_2"/>
<font color="Maroon">y2'</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X</font> <a href="rlsub_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlsub_2.html#T6">RLSUB_2:6</a></i>;<br/><b>then reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">y2'</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E4:37_7_2"/>
<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#T22">RLSUB_1:22</a></i>;<br/><a NAME="E5:37_7_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x0</font></span>)</span>
 <b>by </b><i><a class="ref" href="hahnban.html#T13" target="_self">Th13</a>, <a class="ref" href="rlsub_1.html#T21">RLSUB_1:21</a></i>;<br/><a NAME="E6:37_7_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">t</font> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E7:37_7_2"/><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T22">RLSUB_1:22</a></i>;<br/><a NAME="E8:37_7_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T22">RLSUB_1:22</a></i>;<br/><a NAME="E9:37_7_2"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="hahnban.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><i><font color="Green">E132</font></i>: <font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span>

<br/>.= 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>

;<br/><i><font color="Green">E133</font></i>: <font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, <a class="ref" href="hahnban.html#T4" target="_self">Th4</a></i>
;<br/><b>reconsider </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span></span>)</span></span>)</span> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D4">XXREAL_0:def 4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E13:37_7_2"/>
<span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/><a NAME="E14:37_7_2"/><b>then </b>
 <a href="supinf_1.html#K7">inf</a> <font color="Maroon">B</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T79">SUPINF_1:79</a></i>;<br/><a NAME="E15:37_7_2"/><b>then </b>
 <a href="supinf_1.html#K6">sup</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T8" target="_self">Th8</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:37_7_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E17:37_7_2"/><b>then </b>
<font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E18:37_7_2"/><b>then </b><i><font color="Green">E134</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</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"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><i><font color="Green">E135</font></i>:  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban.html#T4" target="_self">Th4</a></i>
<br/>.= 
<font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T224">XCMPLX_1:224</a></i>
<br/>.= 
<font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span>

;<br/><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> = 
<font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T224">XCMPLX_1:224</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_4.html#T6">RLVECT_4:6</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>
;<br/><a NAME="E21:37_7_2"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T15" target="_self">Th15</a>, </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y2'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="hahnban.html#T13" target="_self">Th13</a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">t</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><a NAME="E23:37_7_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E24:37_7_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E25:37_7_2"/>
<font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="hahnban.html#T14" target="_self">Th14</a>, , , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_7_3"><b>suppose </b></a><a NAME="E1:37_7_3"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:37_7_3"/><b>then </b><i><font color="Green">E137</font></i>: 
 <a href="real_1.html#K1">-</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:37_7_3"/><i><font color="Green">E138</font></i>: 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">y2'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="rlvect_4.html#T6">RLVECT_4:6</a></i>;<br/><a NAME="E4:37_7_3"/><i><font color="Green">E139</font></i>: 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">y2'</font> <a href="rlvect_1.html#R1">in</a>  <a href="rlvect_3.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#T30">RLSUB_1:30</a></i>;<br/><a NAME="E5:37_7_3"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">y2'</font></span>)</span>
 <b>by </b><i><a class="ref" href="hahnban.html#T13" target="_self">Th13</a>, <a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>;<br/><a NAME="E6:37_7_3"/>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">y2'</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">X</font> <a href="rlsub_2.html#K1">+</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlsub_2.html#T6">RLSUB_2:6</a></i>;<br/><b>then reconsider </b><font color="Maroon">y2</font> =  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">y2'</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="rlsub_2.html#K1">+</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1">Lin</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E8:37_7_3"/>
<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T14" target="_self">Th14</a>, <a class="ref" href="rlsub_1.html#T22">RLSUB_1:22</a></i>;<br/><a NAME="E9:37_7_3"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x0</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#T24">RLSUB_1:24</a></i>;<br/><a NAME="E10:37_7_3"/><i><font color="Green">E142</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/><a NAME="E11:37_7_3"/><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T22">RLSUB_1:22</a></i>;<br/><a NAME="E12:37_7_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="rlsub_1.html#T22">RLSUB_1:22</a></i>;<br/><a NAME="E13:37_7_3"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="hahnban.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><i><font color="Green">E145</font></i>: <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span>

<br/>.= 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>

;<br/><i><font color="Green">E146</font></i>: <font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban.html#T10" target="_self">Th10</a>, <a class="ref" href="hahnban.html#T4" target="_self">Th4</a></i>
;<br/><b>reconsider </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D4">XXREAL_0:def 4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E17:37_7_3"/>
<span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="hahnban.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E18:37_7_3"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T76">SUPINF_1:76</a></i>;<br/><a NAME="E19:37_7_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E20:37_7_3"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">y2'</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#T48">RLVECT_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban.html#T14" target="_self">Th14</a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><a NAME="E22:37_7_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
 <b>by </b><i><a class="ref" href="hahnban.html#T16" target="_self">Th16</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E23:37_7_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E24:37_7_3"/>
<font color="Maroon">psi</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>
 <b>by </b><i>, , , , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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