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<b>reconsider </b><font color="Maroon">f</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">V</font>,<font color="Maroon">R</font> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E2:22_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E4:22_1_1"/>
<font color="Maroon">f</font> is    <a href="rmod_4.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">T</font> =  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">V</font> as  <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> ;<br/>
<b>take </b>
<font color="Maroon">T</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rmod_4.html#D4" target="_self">RMOD_4:def 4</a><br/>

<b>thus </b><a NAME="E2:22_1_1_1"/>
for <font color="Olive">v</font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>  st not <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="rmod_4.html#M1" target="_self">Linear_Combination</a> of <font color="Maroon">V</font> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>set </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_1_1_2"/><i><font color="Green">E49</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">{<span class="default"> <font color="Olive">v</font> where <font color="Olive">v</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> </span>}</span> ;<br/><a NAME="E3:22_1_1_2"/>
<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_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E4:22_1_1_2"/><b>then </b>
 ex <font color="Olive">v</font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 ;<br/><b>hence </b><a NAME="E5:22_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:22_1_1"/>
 <a href="rmod_4.html#K2" target="_self">Carrier</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


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