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<b>let </b><font color="Maroon">R</font> be   <a href="vectsp_2.html#NM2">domRing</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM7">RightMod</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>







<a NAME="E1:16"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E2:16"/><b>then </b>
(  <a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">A</font> is    <a href="rmod_2.html#M1">Submodule</a> of  <a href="rmod_5.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> &amp;  <a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">B</font> is    <a href="rmod_2.html#M1">Submodule</a> of  <a href="rmod_5.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:16"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">A</font></span>)</span> <a href="rmod_3.html#K1">+</a> <span class="p1">(<span class="default"><a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">B</font></span>)</span> is    <a href="rmod_2.html#M1">Submodule</a> of  <a href="rmod_5.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rmod_3.html#T40">RMOD_3:40</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:16_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a>  <a href="rmod_5.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">l</font> being    <a href="rmod_4.html#M2">Linear_Combination</a> of <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:16_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="rmod_4.html#K5">Sum</a> <font color="Maroon">l</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">C</font> = <span class="p1">(<span class="default"><a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>;<br/><b>set </b><font color="Maroon">D</font> = <span class="p1">(<span class="default"><a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>;<br/><a NAME="E4:16_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <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">V</font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">H<sub>1</sub></font>(<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">R</font> ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">H<sub>2</sub></font>(<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">R</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_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>assume </b><a NAME="E1:16_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>
 ;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">x</font> as   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> ;<br/><a NAME="E3:16_1_1_1"/>
for <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of 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> holds  <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/><b>hence </b><a NAME="E4:16_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> implies <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <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">R</font> )
 ;<br/><b>assume </b><a NAME="E5:16_1_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 ;<br/><b>thus </b><a NAME="E6:16_1_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:16_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <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">V</font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">H<sub>1</sub></font>(<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">R</font> ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">H<sub>2</sub></font>(<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">R</font> ) )
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of 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> such that </b><br/><a NAME="E6:16_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <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">V</font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S5">FUNCT_2:sch 5</a>();<br/></i><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/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font>;<br/><a NAME="E8:16_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <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">V</font> holds <br/>( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">H<sub>1</sub></font>(<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">R</font> ) &amp; (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">H<sub>2</sub></font>(<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">R</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_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>assume </b><a NAME="E1:16_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">V</font>
 ;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">x</font> as   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> ;<br/><a NAME="E3:16_1_2_1"/>
for <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of 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> holds  <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/><b>hence </b><a NAME="E4:16_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> implies <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <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">R</font> )
 ;<br/><b>assume </b><a NAME="E5:16_1_2_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font>
 ;<br/><b>thus </b><a NAME="E6:16_1_2_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:16_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <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">V</font> holds <br/>( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">H<sub>1</sub></font>(<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">R</font> ) &amp; (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">H<sub>2</sub></font>(<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">R</font> ) )
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of 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> such that </b><br/><a NAME="E10:16_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <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">V</font> holds <br/>( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) &amp; (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S5">FUNCT_2:sch 5</a>();<br/></i><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</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/><b>reconsider </b><font color="Maroon">C</font> = <span class="p1">(<span class="default"><a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> ;<br/><a NAME="E13:16_1"/>
for <font color="Olive">u</font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>  st not <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</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/><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="rmod_4.html#M1">Linear_Combination</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rmod_4.html#D4">RMOD_4:def 4</a></i>;<br/><a NAME="E15:16_1"/><i><font color="Green">E37</font></i>: 
 <a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3"><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="E1:16_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:16_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</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">u</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="E3:16_1_3"/><b>then </b><i><font color="Green">E38</font></i>: 
 ex <font color="Olive">u</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">u</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</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>assume </b><a NAME="E4:16_1_3"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>hence </b><a NAME="E5:16_1_3"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E16:16_1"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E17:16_1"/><b>then </b>
 <a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="rmod_4.html#M2">Linear_Combination</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="rmod_4.html#D7">RMOD_4:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">D</font> = <span class="p1">(<span class="default"><a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> ;<br/><a NAME="E20:16_1"/>
for <font color="Olive">u</font> being  <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>  st not <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</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/><b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as    <a href="rmod_4.html#M1">Linear_Combination</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rmod_4.html#D4">RMOD_4:def 4</a></i>;<br/><a NAME="E22:16_1"/><i><font color="Green">E58</font></i>: 
 <a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_4"><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="E1:16_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:16_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font> : <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</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="E3:16_1_4"/><b>then </b><i><font color="Green">E59</font></i>: 
 ex <font color="Olive">u</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">u</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">u</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>assume </b><a NAME="E4:16_1_4"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>hence </b><a NAME="E5:16_1_4"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E23:16_1"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_5"><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="E1:16_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:16_1_5"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  <a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="rmod_4.html#D7">RMOD_4:def 7</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:16_1"/><b>then </b>
 <a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as    <a href="rmod_4.html#M2">Linear_Combination</a> of <font color="Maroon">B</font> <b>by </b><i><a class="ref" href="rmod_4.html#D7">RMOD_4:def 7</a></i>;<br/><a NAME="E26:16_1"/>
<font color="Maroon">l</font> <a href="rmod_4.html#R1">=</a> <font color="Maroon">f</font> <a href="rmod_4.html#K6">+</a> <font color="Maroon">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rmod_4.html#D10">RMOD_4:def 10</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_6_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_6_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_6_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_6_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/><a NAME="E2:16_1_6_1_1_1_1"/><b>then </b>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:16_1_6_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rmod_4.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rmod_4.html#D11">RMOD_4:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_6_1_1_2"/><i><font color="Green">E62</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_6_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_6_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_6_1_1_2_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font> or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_6_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_6_1_1_2_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font>
 ;<br/><div class="add"><div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_6_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_6_1_1_2_1_1_1_1"/>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/><a NAME="E2:16_1_6_1_1_2_1_1_1_1"/><b>then </b>
(  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font> or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:16_1_6_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rmod_4.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rmod_4.html#D11">RMOD_4:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_6_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_6_1_1_2_1_1_2"/><i><font color="Green">E65</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="rmod_4.html#K2">Carrier</a> <font color="Maroon">l</font>
 ;<br/><div class="add"><a NAME="E2:16_1_6_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
( not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rmod_4.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rmod_4.html#D11">RMOD_4:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:16_1_6_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rmod_4.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:16_1_6"/>
<font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rmod_4.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E27:16_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K5">Sum</a> <font color="Maroon">f</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rmod_4.html#K5">Sum</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rmod_4.html#T77">RMOD_4:77</a></i>;<br/><a NAME="E28:16_1"/>
(  <a href="rmod_4.html#K5">Sum</a> <font color="Maroon">f</font> <a href="rlvect_1.html#R1">in</a>  <a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">A</font> &amp;  <a href="rmod_4.html#K5">Sum</a> <font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a>  <a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E29:16_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">A</font></span>)</span> <a href="rmod_3.html#K1">+</a> <span class="p1">(<span class="default"><a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rmod_3.html#T5">RMOD_3:5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:16"/><b>then </b>
 <a href="rmod_5.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> is    <a href="rmod_2.html#M1">Submodule</a> of <span class="p1">(<span class="default"><a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">A</font></span>)</span> <a href="rmod_3.html#K1">+</a> <span class="p1">(<span class="default"><a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rmod_2.html#T36">RMOD_2:36</a></i>;<br/>
<b>hence </b><a NAME="E6:16"/>
 <a href="rmod_5.html#K1" target="_self">Lin</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">A</font></span>)</span> <a href="rmod_3.html#K1">+</a> <span class="p1">(<span class="default"><a href="rmod_5.html#K1" target="_self">Lin</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rmod_2.html#T33">RMOD_2:33</a></i>;<br/>


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