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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="rusub_4.html#V2">Affine</a> <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="E1:16"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="rusub_4.html#V3">Subspace-like</a>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="rusub_4.html#V3">Subspace-like</a>
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rusub_5.html#R1" target="_self">is_parallel_to</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:16"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rusub_5.html#R1" target="_self">is_parallel_to</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E5:16"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rusub_5.html#R1" target="_self">is_parallel_to</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E3:16"><i><font color="Green">E12</font></i></a>, <a class="ref" href="rusub_5.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E6:16"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="rusub_5.html#R1" target="_self">is_parallel_to</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E4:16"><i><font color="Green">E13</font></i></a>, <a class="ref" href="rusub_5.html#T3" target="_self">Th3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</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="E8:16"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="rusub_4.html#K7">+</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="rusub_5.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E9:16"/>
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E1:16"><i><font color="Green">E10</font></i></a>, <a class="ref" href="rusub_4.html#D8">RUSUB_4:def 8</a></i>;<br/>
<a NAME="E10:16"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E11:16"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="rusub_5.html#E8:16"><i><font color="Green">E14</font></i></a>, <a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</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="E13:16"/><i><font color="Green">E15</font></i>: 
(  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>
<a NAME="E14:16"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E13:16"><i><font color="Green">E15</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:16"/><b>then </b><i><font color="Green">E16</font></i>: 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E13:16"><i><font color="Green">E15</font></i></a>, <a class="ref" href="rlvect_1.html#T19">RLVECT_1:19</a></i>;<br/>
<font color="Maroon">c<sub>5</sub></font> = 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</a></i>
;<br/>
<a NAME="E17:16"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E2:16"><i><font color="Green">E11</font></i></a>, <a class="txt" href="rusub_5.html#E15:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="rusub_4.html#D8">RUSUB_4:def 8</a></i>;<br/>
<a NAME="E18:16"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E2:16"><i><font color="Green">E11</font></i></a>, <a class="txt" href="rusub_5.html#E8:16"><i><font color="Green">E14</font></i></a>, <a class="ref" href="rusub_5.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E19:16"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rusub_5.html#R1" target="_self">is_parallel_to</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E4:16"><i><font color="Green">E13</font></i></a>, <a class="ref" href="rusub_5.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E20:16"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="rusub_5.html#R1" target="_self">is_parallel_to</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E3:16"><i><font color="Green">E12</font></i></a>, <a class="ref" href="rusub_5.html#T3" target="_self">Th3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</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="E22:16"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="rusub_4.html#K7">+</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="rusub_5.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E23:16"/>
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E2:16"><i><font color="Green">E11</font></i></a>, <a class="ref" href="rusub_4.html#D8">RUSUB_4:def 8</a></i>;<br/>
<a NAME="E24:16"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<a NAME="E25:16"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="rusub_5.html#E22:16"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</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="E27:16"/><i><font color="Green">E19</font></i>: 
(  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>
<a NAME="E28:16"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E27:16"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E29:16"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E27:16"><i><font color="Green">E19</font></i></a>, <a class="ref" href="rlvect_1.html#T19">RLVECT_1:19</a></i>;<br/>
<font color="Maroon">c<sub>8</sub></font> = 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</a></i>
;<br/>
<a NAME="E31:16"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E1:16"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rusub_5.html#E29:16"><i><font color="Green">E20</font></i></a>, <a class="ref" href="rusub_4.html#D8">RUSUB_4:def 8</a></i>;<br/>
<a NAME="E32:16"/><b>then </b>
<font color="Maroon">c<sub>4</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="rusub_5.html#E1:16"><i><font color="Green">E10</font></i></a>, <a class="txt" href="rusub_5.html#E22:16"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rusub_5.html#T13" target="_self">Th13</a></i>;<br/>
<b>hence </b><a NAME="E33:16"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="rusub_5.html#E18:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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