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

<b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a> <a href="bhsp_1.html#V2">RealUnitarySpace-like</a>  <a href="bhsp_1.html#L1">UNITSTR</a> ;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>







<b>assume </b><a NAME="E1:3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font></span>)</span> holds <br/> ex <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">u</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) </span>}</span> 
 ;<br/>

<b>let </b><font color="Maroon">u1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v1</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="convex1.html#D2">CONVEX1:def 2</a><br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:3"/><i><font color="Green">E37</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 <b>and </b><br/><a NAME="E3:3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 ;<br/>

<b>consider </b><font color="Maroon">u'</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font>
 <b>and </b><br/><a NAME="E6:3"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font></span>)</span> holds <br/> ex <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">u'</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">v'</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E8:3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font>
 <b>and </b><br/><a NAME="E9:3"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font></span>)</span> holds <br/> ex <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">v'</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E10:3"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font></span>)</span> holds <br/> ex <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Olive">v</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:3_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<a NAME="E2:3_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:3_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">u'</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E7:3_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">v'</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/>
<b>take </b><font color="Maroon">v</font> = <font color="Maroon">x</font>;<br/>

<a NAME="E8:3_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E9:3_1"/>
0 <a href="real_1.html#K3">+</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E10:3_1"/><b>then </b>
1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E11:3_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_1.html#D2">BHSP_1:def 2</a></i>
;<br/>
<a NAME="E13:3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E14:3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E15:3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>hence </b><a NAME="E17:3_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="bhsp_1.html#K1">.|.</a> <font color="Maroon">v</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:3"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>


</div>
