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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/>

<b>assume </b><a NAME="E1:31"/><i><font color="Green">E32</font></i>: 
 ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>for <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">w</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font></span>)</span>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font></span>)</span>, <font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font></span>)</span>;<br/>

<b>set </b><font color="Maroon">S</font> =  <a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font>;<br/>
<b>assume </b><a NAME="E2:31"/><i><font color="Green">E36</font></i>: 
( not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1" target="_self">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font></span>)</span> &amp; not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1" target="_self">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font></span>)</span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="analoaf.html#D2" target="_self">ANALOAF:def 2</a><br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font>, <font color="Maroon">c'</font> = <font color="Maroon">c</font>, <font color="Maroon">d'</font> = <font color="Maroon">d</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E4:31"/>
( not <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">c'</font>,<font color="Maroon">d'</font> &amp; not <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">d'</font>,<font color="Maroon">c'</font> )
 
<b>by </b><i><a class="txt" href="analoaf.html#E1:31"><i><font color="Green">E32</font></i></a>, <a class="txt" href="analoaf.html#E3"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">t'</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:31"/><i><font color="Green">E41</font></i>: 
( ( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">a'</font>,<font color="Maroon">t'</font> or <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">t'</font>,<font color="Maroon">a'</font> ) &amp; ( <font color="Maroon">c'</font>,<font color="Maroon">d'</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">c'</font>,<font color="Maroon">t'</font> or <font color="Maroon">c'</font>,<font color="Maroon">d'</font> <a href="analoaf.html#R1" target="_self">//</a> <font color="Maroon">t'</font>,<font color="Maroon">c'</font> ) )
 <b>by </b><i><a class="ref" href="analoaf.html#T9" target="_self">Th9</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font></span>)</span> ;<br/>
<a NAME="E8:31"/>
( ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1" target="_self">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font></span>)</span> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1" target="_self">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font></span>)</span> ) &amp; ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1" target="_self">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font></span>)</span> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1" target="_self">CONGR</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="analoaf.html#T10" target="_self">Th10</a>, <a class="txt" href="analoaf.html#E3"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E9:31"/><b>then </b>
( ( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Maroon">a</font>,<font color="Maroon">t</font> or <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Maroon">t</font>,<font color="Maroon">a</font> ) &amp; ( <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Maroon">c</font>,<font color="Maroon">t</font> or <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Maroon">t</font>,<font color="Maroon">c</font> ) )
 
<b>by </b><i><a class="ref" href="analoaf.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E10:31"/>
 ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font></span>)</span> st <br/>( ( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Maroon">a</font>,<font color="Olive">t</font> or <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">t</font>,<font color="Maroon">a</font> ) &amp; ( <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Maroon">c</font>,<font color="Olive">t</font> or <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">t</font>,<font color="Maroon">c</font> ) )
 ;<br/>


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