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

<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font><br/> for <font color="Olive" title="b2">W1</font>, <font color="Olive" title="b3">W2</font>, <font color="Olive" title="b4">W3</font>, <font color="Olive" title="b5">W4</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V</font>  st <font color="Olive" title="b2">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b3">W2</font> &amp; <font color="Olive" title="b4">W3</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b5">W4</font> &amp;  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b2">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b4">W3</font> &amp;  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b3">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b5">W4</font> holds <br/> <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Olive" title="b2">W1</font>,<font color="Olive" title="b3">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Olive" title="b4">W3</font>,<font color="Olive" title="b5">W4</font></span><br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font>, <font color="Olive" title="b3">W3</font>, <font color="Olive" title="b4">W4</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b1">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b2">W2</font> &amp; <font color="Olive" title="b3">W3</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b4">W4</font> &amp;  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b1">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b3">W3</font> &amp;  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Olive" title="b4">W4</font> holds <br/> <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Olive" title="b1">W1</font>,<font color="Olive" title="b2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Olive" title="b3">W3</font>,<font color="Olive" title="b4">W4</font></span><br/><b>let </b><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font>, <font color="Maroon" title="c5">W3</font>, <font color="Maroon" title="c6">W4</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c5">W3</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">W4</font> &amp;  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c5">W3</font> &amp;  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c6">W4</font> implies  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c5">W3</font>,<font color="Maroon" title="c6">W4</font> )</span><br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c3">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c5">W3</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">W4</font> &amp;  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c5">W3</font> &amp;  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c6">W4</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c5">W3</font>,<font color="Maroon" title="c6">W4</font></span><br/>

<b>thus </b><a NAME="E2:8"/>
 <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c5">W3</font>,<font color="Maroon" title="c6">W4</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c5">W3</font>,<font color="Maroon" title="c6">W4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span><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" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font> or <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c5">W3</font>,<font color="Maroon" title="c6">W4</font> )</span><br/>

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c5">W3</font>,<font color="Maroon" title="c6">W4</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">A1</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" title="VECTSP_5:def.3">VECTSP_5:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">A</font> = <font color="Maroon" title="c7">a</font> as  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E3:8_1"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/>
<a NAME="E5:8_1"/>
( <font color="Maroon" title="c3">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c9">A</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">W2</font> )
 
<b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E1:8_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="pencil_4.html#D1" target="_self" title="PENCIL_4:def.1">Def1</a></i>;<br/>
<a NAME="E6:8_1"/><b>then </b>
(  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c9">A</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c4">W2</font> )
 
<b>by </b><i><a class="ref" href="pencil_4.html#T3" target="_self" title="PENCIL_4:th.3">Th3</a>, <a class="ref" href="pencil_4.html#T5" target="_self" title="PENCIL_4:th.5">Th5</a></i>;<br/>
<a NAME="E7:8_1"/><b>then </b>
( <font color="Maroon" title="c5">W3</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c9">A</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">W4</font> )
 
<b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="pencil_4.html#T4" target="_self" title="PENCIL_4:th.4">Th4</a>, <a class="ref" href="pencil_4.html#T6" target="_self" title="PENCIL_4:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E8:8_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c5">W3</font>,<font color="Maroon" title="c6">W4</font>
 <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="pencil_4.html#D1" target="_self" title="PENCIL_4:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c5">W3</font>,<font color="Maroon" title="c6">W4</font> or <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font> )</span><br/>

<b>assume </b><a NAME="E3:8"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c5">W3</font>,<font color="Maroon" title="c6">W4</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">A1</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E5:8"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" title="VECTSP_5:def.3">VECTSP_5:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">A</font> = <font color="Maroon" title="c7">a</font> as  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E5:8"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>
<a NAME="E7:8"/>
( <font color="Maroon" title="c5">W3</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c9">A</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">W4</font> )
 
<b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E3:8"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="pencil_4.html#D1" target="_self" title="PENCIL_4:def.1">Def1</a></i>;<br/>
<a NAME="E8:8"/><b>then </b>
(  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c5">W3</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c9">A</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c6">W4</font> )
 
<b>by </b><i><a class="ref" href="pencil_4.html#T3" target="_self" title="PENCIL_4:th.3">Th3</a>, <a class="ref" href="pencil_4.html#T5" target="_self" title="PENCIL_4:th.5">Th5</a></i>;<br/>
<a NAME="E9:8"/><b>then </b>
( <font color="Maroon" title="c3">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c9">A</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">W2</font> )
 
<b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="pencil_4.html#T4" target="_self" title="PENCIL_4:th.4">Th4</a>, <a class="ref" href="pencil_4.html#T6" target="_self" title="PENCIL_4:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E10:8"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K1" title="PENCIL_4:func.1">segment</a> <font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>
 <b>by </b><i><a class="txt" href="#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="pencil_4.html#D1" target="_self" title="PENCIL_4:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
