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<b>consider </b><font color="Maroon">V0</font> being  non <a href="realset2.html#V3">trivial</a> <a href="rlvect_1.html#NM2">RealLinearSpace</a><b> such that </b><br/><a NAME="E2:61_1_1"/><i><font color="Green">E58</font></i>: 
 ex <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V0</font> st <br/>( ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a>  st <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">w</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V0</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; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> 0 ) ) &amp; ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V0</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">w</font></span>)</span> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">u0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V0</font>, <font color="Maroon">v0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V0</font>, <font color="Maroon">w0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V0</font><b> such that </b><br/><a NAME="E4:61_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a>  st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u0</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V0</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; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> 0 )
 <b>and </b><br/><a NAME="E5:61_1_1"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V0</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u0</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">c</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w0</font></span>)</span>
 <b>by </b><i><a class="ref" href="anproj_2.html#T1" target="_self">Th1</a></i>;<br/>
<b>take </b>
<font color="Maroon">V0</font>
;<br/>

<b>thus </b><a NAME="E6:61_1_1"/>
<font color="Maroon">V0</font> is <a href="anproj_2.html#V1" target="_self">up-3-dimensional</a>
 <b>by </b><i><a class="txt" href="anproj_2.html#E2:61_1_1"><i><font color="Green">E58</font></i></a>, </i>;<br/>


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