<?xml version="1.0"?>
<div class="add">

<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="rlvect_1.html#NM2">RealLinearSpace</a>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:33_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>and </b><br/><a NAME="E3:33_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></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">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></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">c<sub>1</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; ( <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font> implies <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> implies <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> implies <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> ) &amp; (  not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font> ) ) ) &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font></span>)</span>ex <font color="Olive">b<sub>4</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>4</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font> holds <br/>ex <font color="Olive">b<sub>5</sub></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">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="analoaf.html#R2" target="_self">//</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></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#T35" target="_self">Th35</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">c<sub>1</sub></font> is <a href="realset2.html#V3">trivial</a> &amp;  <a href="analoaf.html#K2" target="_self">OASpace</a> <font color="Maroon">c<sub>1</sub></font> is <a href="analoaf.html#V2" target="_self">OAffinSpace-like</a> )
 
<b>by </b><i><a class="ref" href="analoaf.html#D5" target="_self">Def5</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/>


</div>
