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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E2:8"/>
 <a href="pencil_4.html#K1" target="_self">segment</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pencil_4.html#K1" target="_self">segment</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="pencil_4.html#K1" target="_self">segment</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3">VECTSP_5:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pencil_4.html#E3:8_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E5:8_1"/>
( <font color="Maroon">c<sub>3</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="pencil_4.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="pencil_4.html#E1:8_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="pencil_4.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E6:8_1"/><b>then </b>
(  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="pencil_4.html#T3" target="_self">Th3</a>, <a class="ref" href="pencil_4.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E7:8_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="pencil_4.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="ref" href="pencil_4.html#T6" target="_self">Th6</a>, <a class="ref" href="pencil_4.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E8:8_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="pencil_4.html#K1" target="_self">segment</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="ref" href="pencil_4.html#D1" target="_self">Def1</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:8"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="pencil_4.html#K1" target="_self">segment</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:8"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3">VECTSP_5:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pencil_4.html#E5:8"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E7:8"/>
( <font color="Maroon">c<sub>5</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="pencil_4.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="txt" href="pencil_4.html#E3:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="pencil_4.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E8:8"/><b>then </b>
(  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="pencil_4.html#T3" target="_self">Th3</a>, <a class="ref" href="pencil_4.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E9:8"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="pencil_4.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="ref" href="pencil_4.html#T6" target="_self">Th6</a>, <a class="ref" href="pencil_4.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E10:8"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="pencil_4.html#K1" target="_self">segment</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="ref" href="pencil_4.html#D1" target="_self">Def1</a></i>;<br/>


</div>
