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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="E1:39"/><i><font color="Green">E32</font></i>: 
 ex <font color="Olive">u</font>, <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> st <br/>( ( for <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">u</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">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 ) ) &amp; ( 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 <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">u</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">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>consider </b><font color="Maroon">u</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:39"/><i><font color="Green">E36</font></i>: 
for <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="Maroon">u</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="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 )
 <b>and </b><br/><a NAME="E4:39"/><i><font color="Green">E41</font></i>: 
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 <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</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="Maroon">v</font></span>)</span>
 <b>by </b><i><a class="ref" href="analoaf.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E5:39"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</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 not <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">c</font>,<font color="Olive">d</font> &amp; not <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">d</font>,<font color="Olive">c</font> holds <br/> ex <font color="Olive">p</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="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">a</font>,<font color="Olive">p</font> or <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">p</font>,<font color="Olive">a</font> ) &amp; ( <font color="Olive">c</font>,<font color="Olive">d</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">c</font>,<font color="Olive">p</font> or <font color="Olive">c</font>,<font color="Olive">d</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">p</font>,<font color="Olive">c</font> ) )
 
<b>by </b><i><a class="txt" href="analoaf.html#E3:39"><i><font color="Green">E36</font></i></a>, <a class="ref" href="analoaf.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E6:39"/>
 <a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font> is   <a href="analoaf.html#NM2">OAffinPlane</a>
 <b>by </b><i><a class="ref" href="analoaf.html#T12" target="_self">Th12</a>, <a class="ref" href="analoaf.html#T10" target="_self">Th10</a>, <a class="txt" href="analoaf.html#E1:39"><i><font color="Green">E32</font></i></a></i>;<br/>


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