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

<b>consider </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">x3</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">x4</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E2:59"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K9">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>,<font color="Maroon">x4</font></span><a href="domain_1.html#K9">}</a></span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x4</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x4</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x4</font> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">V</font> =  <a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font>;<br/>
<b>consider </b><font color="Maroon">f</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> , <font color="Maroon">g</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> , <font color="Maroon">h</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> , <font color="Maroon">f1</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:59"/><i><font color="Green">E59</font></i>: 
( ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a>  st <span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">h</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">d</font>,<font color="Maroon">f1</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="funcsdom.html#K9">RealFuncZero</a> <font color="Maroon">A</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">d</font> <a href="hidden.html#R1">=</a> 0 ) ) &amp; ( for <font color="Olive">h'</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a>   ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">h'</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">h</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">d</font>,<font color="Maroon">f1</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> ) )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">f</font>, <font color="Maroon">v</font> = <font color="Maroon">g</font>, <font color="Maroon">w</font> = <font color="Maroon">h</font>, <font color="Maroon">u1</font> = <font color="Maroon">f1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<a NAME="E6:59"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">d</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font></span>)</span> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">d</font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">t</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:59_1"/><i><font color="Green">E61</font></i>: 
<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">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <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">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font>, <font color="Maroon">v'</font> = <font color="Maroon">s</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>, <font color="Maroon">w'</font> = <font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>, <font color="Maroon">u1'</font> = <font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">u''</font> = <font color="Maroon">u'</font>, <font color="Maroon">v''</font> = <font color="Maroon">v'</font>, <font color="Maroon">w''</font> = <font color="Maroon">w'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font></span>)</span> ;<br/>
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font></span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u''</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v''</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w''</font></span>)</span>,<font color="Maroon">u1'</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="anproj_2.html#E4:59"><i><font color="Green">E59</font></i></a>, <a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p3">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">u''</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v''</font></span>)</span>,<font color="Maroon">w'</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">u1'</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">u'</font>,<font color="Maroon">v'</font></span>)</span>,<font color="Maroon">w'</font></span>)</span>,<font color="Maroon">u1'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">v'</font></span>)</span>,<font color="Maroon">w'</font></span>)</span>,<font color="Maroon">u1'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<font color="Maroon">w'</font></span>)</span>,<font color="Maroon">u1'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">h</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<font color="Maroon">u1'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">h</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">f1</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
;<br/>
<a NAME="E5:59_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">h</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">f1</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="funcsdom.html#K9">RealFuncZero</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:59_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:59"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font></span>)</span>  ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">d</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">h'</font> = <font color="Maroon">y</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a>  ;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:59_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">h'</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">h</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">f1</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:59_2"/>
( <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">h</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">d</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">f1</font></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>;<br/>
<b>then </b><font color="Maroon">h'</font> = 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p3">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="anproj_2.html#E6:59"><i><font color="Green">E60</font></i></a>, <a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E6:59_2"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">d</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:59"/>
<font color="Maroon">u</font> <a href="anproj_1.html#NR1" target="_self">is_Prop_Vect</a> 
 
<b>by </b><i><a class="txt" href="anproj_2.html#E2:59"><i><font color="Green">E58</font></i></a>, </i>;<br/>
<a NAME="E9:59"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">V</font> =  <a href="funcsdom.html#K11">RealVectSpace</a> <font color="Maroon">A</font> as  non <a href="realset2.html#V3">trivial</a> <a href="rlvect_1.html#NM2">RealLinearSpace</a> <b>by </b><i><a class="ref" href="anproj_1.html#D8">ANPROJ_1:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">V</font>
;<br/>

<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font>, <font color="Maroon">v</font> = <font color="Maroon">v</font>, <font color="Maroon">w</font> = <font color="Maroon">w</font>, <font color="Maroon">u1</font> = <font color="Maroon">u1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

<b>take </b>
<font color="Maroon">v</font>
;<br/>

<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>take </b>
<font color="Maroon">u1</font>
;<br/>

<b>thus </b><a NAME="E12:59"/>
( ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">d</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">d</font> <a href="hidden.html#R1">=</a> 0 ) ) &amp; ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">d</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:59"><i><font color="Green">E58</font></i></a>, <a class="txt" href="anproj_2.html#E4:59"><i><font color="Green">E59</font></i></a></i>;<br/>


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