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<div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be  <a href="collsp.html#V4" target="_self">proper</a> <a href="collsp.html#NM2">CollSp</a>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>10</sub></font>;<br/>





<b>assume </b><a NAME="E1:30"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<a NAME="E2:30"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="collsp.html#K1" target="_self">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:30_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="collsp.html#K1" target="_self">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> implies <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="collsp.html#K1" target="_self">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<a NAME="E2:30_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1" target="_self">is_collinear</a>  </span>}</span> 
 
;<br/>
<a NAME="E3:30_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>10</sub></font> st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1" target="_self">is_collinear</a>  )
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>10</sub></font> ;<br/>
<a NAME="E5:30_1_1"/>
<font color="Maroon">c<sub>14</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>
 
;<br/>
<b>hence </b><a NAME="E6:30_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:30_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font> implies <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="collsp.html#K1" target="_self">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>10</sub></font> ;<br/>
<a NAME="E3:30_1_2"/>
<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="collsp.html#R1" target="_self">is_collinear</a> 
 
<b>by </b><i><a class="txt" href="collsp.html#E1:30"><i><font color="Green">E25</font></i></a>, <a class="ref" href="collsp.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E4:30_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="collsp.html#R1" target="_self">is_collinear</a>  </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E5:30_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="collsp.html#K1" target="_self">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30"/>
 <a href="collsp.html#K1" target="_self">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
