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

<b>let </b><font color="Maroon">V1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">K</font>, <font color="Maroon">V2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">K</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:43_1_2"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V1</font> iff <font color="Olive">x</font> is   <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">V</font> )
 <b>and </b><br/><a NAME="E2:43_1_2"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">V</font> holds  the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="hahnban1.html#K3" target="_self">+</a> <font color="Olive">g</font>
 <b>and </b><br/><a NAME="E3:43_1_2"/><i><font color="Green">E90</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V1</font> <a href="hidden.html#R1">=</a>  <a href="hahnban1.html#K7" target="_self">0Functional</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E4:43_1_2"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">f</font> being  <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">V</font><br/> for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> holds  the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Olive">f</font>
 <b>and </b><br/><a NAME="E5:43_1_2"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V2</font> iff <font color="Olive">x</font> is   <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">V</font> )
 <b>and </b><br/><a NAME="E6:43_1_2"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">V</font> holds  the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="hahnban1.html#K3" target="_self">+</a> <font color="Olive">g</font>
 <b>and </b><br/><a NAME="E7:43_1_2"/><i><font color="Green">E94</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V2</font> <a href="hidden.html#R1">=</a>  <a href="hahnban1.html#K7" target="_self">0Functional</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E8:43_1_2"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">f</font> being  <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">V</font><br/> for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> holds  the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Olive">f</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:43_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V1</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V1</font>
 ;<br/>

<a NAME="E2:43_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> is   <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:43_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:43_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V2</font>
 ;<br/><a NAME="E3:43_1_2_1"/><b>then </b>
<font color="Maroon">x</font> is   <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:43_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E10:43_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V1</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V1</font>;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font>, <font color="Maroon">g</font> = <font color="Maroon">y</font> as   <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/><b>thus </b>the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">f</font> <a href="hahnban1.html#K3" target="_self">+</a> <font color="Maroon">g</font>
<b>by </b><i/>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<a NAME="E12:43_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V1</font> <a href="funct_2.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V2</font>
 
<b>by </b><i><a class="ref" href="hahnban1.html#T5" target="_self">Th5</a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V1</font>;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as   <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/><b>thus </b>the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">r</font>,<font color="Maroon">x</font> = 
<font color="Maroon">r</font> <a href="hahnban1.html#K6" target="_self">*</a> <font color="Maroon">f</font>
<b>by </b><i/>
<br/>.= 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:43_1_2"/>
<font color="Maroon">V1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V2</font>
 <b>by </b><i>, , <a class="ref" href="hahnban1.html#T5" target="_self">Th5</a>, , <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div>
