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<b>consider </b><font color="Maroon">V</font> being   <a href="rlvect_1.html#NM2">RealLinearSpace</a>, <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="E2:33_1_1"/><i><font color="Green">E32</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="E3:33_1_1"/>
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="funcsdom.html#T37">FUNCSDOM:37</a></i>;<br/>
<a NAME="E4:33_1_1"/>
(  ex <font color="Olive">a</font>, <font color="Olive">b</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 <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font>, <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font>, <font color="Olive">s</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> holds <br/> ( <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">c</font> &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b</font>,<font color="Olive">a</font> implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> ) &amp; ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <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">q</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">r</font>,<font color="Olive">s</font> implies <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">r</font>,<font color="Olive">s</font> ) &amp; ( <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> implies <font color="Olive">b</font>,<font color="Olive">a</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">d</font>,<font color="Olive">c</font> ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b</font>,<font color="Olive">c</font> implies <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">c</font> ) &amp; (  not <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">c</font> or <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b</font>,<font color="Olive">c</font> or <font color="Olive">a</font>,<font color="Olive">c</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">c</font>,<font color="Olive">b</font> ) ) ) &amp;  ex <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 <br/>( 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> ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</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>  ex <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 <br/>( <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; <font color="Olive">a</font>,<font color="Olive">c</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b</font>,<font color="Olive">d</font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</font> ) ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</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 <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font>,<font color="Olive">p</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">p</font>,<font color="Olive">c</font> holds <br/> ex <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 <br/>( <font color="Olive">a</font>,<font color="Olive">p</font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">p</font>,<font color="Olive">d</font> &amp; <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> ) ) )
 
<b>by </b><i><a class="txt" href="analoaf.html#E2:33_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="analoaf.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:33_1_1"/><b>then </b>
( not  <a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font> is <a href="realset2.html#V3">trivial</a> &amp;  <a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">V</font> is <a href="analoaf.html#V2" target="_self">OAffinSpace-like</a> )
 
<b>by </b><i><a class="ref" href="analoaf.html#T9" target="_self">Th9</a>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:33_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="analoaf.html#L1" target="_self">AffinStruct</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="analoaf.html#V1" target="_self">strict</a> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="realset2.html#V3">trivial</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="analoaf.html#V2" target="_self">OAffinSpace-like</a> )
 ;<br/>


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