<?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="vectsp_1.html#NM6">VectSp</a> of  <a href="complfld.html#K1">F_Complex</a> ;<br/><b>let </b><font color="Maroon">M</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/>



<a NAME="E1:69"/><i><font color="Green">E35</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/>
<a NAME="E2:69"/><i><font color="Green">E36</font></i>: 
 <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">M</font> #) <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E3:69"/><i><font color="Green">E37</font></i>: 
 <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E4:69"/><i><font color="Green">E38</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></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"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="txt" href="hahnban1.html#E2:69"><i><font color="Green">E36</font></i></a>, <a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlsub_1.html#D2">RLSUB_1:def 2</a><br/>

<b>thus </b><a NAME="E5:69"/>
the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="txt" href="hahnban1.html#E2:69"><i><font color="Green">E36</font></i></a>, <a class="txt" href="hahnban1.html#E3:69"><i><font color="Green">E37</font></i></a>, <a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/>

<b>thus </b><a NAME="E6:69"/>
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i><a class="txt" href="hahnban1.html#E2:69"><i><font color="Green">E36</font></i></a>, <a class="txt" href="hahnban1.html#E3:69"><i><font color="Green">E37</font></i></a>, <a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/>

<a NAME="E7:69"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="hahnban1.html#E4:69"><i><font color="Green">E38</font></i></a>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E8:69"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:69"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E10:69"/><i><font color="Green">E40</font></i>: 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> <a href="relset_1.html#K8">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/>
<a NAME="E11:69"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</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"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:69_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:69_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:69_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U2">Mult</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:69_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="hahnban1.html#E3:69_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="hahnban1.html#E3:69_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="hahnban1.html#T8" target="_self">Th8</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i><a class="txt" href="hahnban1.html#E9:69"><i><font color="Green">E39</font></i></a>, <a class="txt" href="hahnban1.html#E3:69_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="hahnban1.html#T8" target="_self">Th8</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> <b>by </b><i><a class="txt" href="hahnban1.html#E2:69"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">b2</font> = <font color="Maroon">b1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="txt" href="hahnban1.html#E1:69"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:69_1"/><i><font color="Green">E85</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:69_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E13:69_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:69_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="hahnban1.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E15:69_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> <a href="relset_1.html#K8">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<font color="Maroon">y</font> = 
the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="hahnban1.html#E3:69_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="hahnban1.html#E4:69_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="hahnban1.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b2</font>
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="txt" href="hahnban1.html#E10:69"><i><font color="Green">E40</font></i></a>, <a class="ref" href="hahnban1.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
;<br/>
<b>hence </b><a NAME="E17:69_1"/>
<font color="Maroon">y</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="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">RM</font> = the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i><a class="txt" href="hahnban1.html#E9:69"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> <b>by </b><i><a class="txt" href="hahnban1.html#E2:69"><i><font color="Green">E36</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">b2</font> = <font color="Maroon">b1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="txt" href="hahnban1.html#E1:69"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E3:69_2"/><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:69_2"/>
<font color="Maroon">b</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="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="txt" href="hahnban1.html#E4:69"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:69_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:69_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:69_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> the <a href="rlvect_1.html#U2">Mult</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:69_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">RM</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E9:69_2"/><i><font color="Green">E91</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:69_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E11:69_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E12:69_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="hahnban1.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E13:69_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> <a href="relset_1.html#K8">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><b>thus </b>the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><font color="Maroon">a</font>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b2</font>
<b>by </b><i><a class="txt" href="hahnban1.html#E10:69"><i><font color="Green">E40</font></i></a>, <a class="ref" href="hahnban1.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<font color="Maroon">RM</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:69"/>
the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U2">Mult</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div>
