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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 ;<br/>

<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:53_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:53_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E3:53_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:53"><i><font color="Green">E45</font></i></a>, <a class="txt" href="aff_4.html#E1:53_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="aff_4.html#E1:53_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E4:53_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:53"><i><font color="Green">E45</font></i></a>, <a class="txt" href="aff_4.html#E2:53_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="aff_4.html#T33" target="_self">Th33</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:53_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="aff_4.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="aff_4.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="aff_4.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:53_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:53_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:53"><i><font color="Green">E45</font></i></a>, <a class="ref" href="aff_1.html#T31">AFF_1:31</a></i>;<br/><a NAME="E5:53_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:53"><i><font color="Green">E45</font></i></a>, <a class="txt" href="aff_4.html#E1:53_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="aff_4.html#E3:53_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="aff_4.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E6:53_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E1:53"><i><font color="Green">E45</font></i></a>, <a class="ref" href="aff_4.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E7:53_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="aff_4.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:53"><i><font color="Green">E45</font></i></a>, <a class="txt" href="aff_4.html#E5:53_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="aff_4.html#T33" target="_self">Th33</a></i>;<br/><b>hence </b><a NAME="E8:53_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:53"><i><font color="Green">E45</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:53"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="aff_4.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E2:53"><i><font color="Green">E46</font></i></a></i>;<br/>


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