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

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> : for <font color="Olive">w</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">v</font>,<font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> </span>}</span> ;<br/>
<a NAME="E1:90_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> : for <font color="Olive">w</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">v</font>,<font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:90_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> : for <font color="Olive">w</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">v</font>,<font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:90_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">w</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:90_1_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">V</font>
 <b>by </b><i><a class="ref" href="bilinear.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:90_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> : for <font color="Olive">w</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">W</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">v</font>,<font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>
 ;<br/>


</div>
