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

<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">VectSp</a> of <font color="Maroon" title="c1">F</font>;<br/><b>let </b><font color="Maroon" title="c3">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E34">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c4">S</font> =  <a href="pencil_4.html#K7" title="PENCIL_4:func.7">GrassmannSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c5">K</font>, <font color="Maroon" title="c6">L</font> be   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K7" title="PENCIL_4:func.7">GrassmannSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_1.html#D7" title="PENCIL_1:def.7">PENCIL_1:def 7</a><br/>

<a NAME="E2:35"/><i><font color="Green" title="E35">A2</font></i>: 
not the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K7" title="PENCIL_4:func.7">GrassmannSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="pencil_4.html#T24" target="_self" title="PENCIL_4:th.24">Th24</a>, <a class="txt" href="pencil_4.html#E1:35"><i><font color="Green" title="E34">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">W1</font> being    <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="E4:35"/><i><font color="Green" title="E36">A3</font></i>: 
(  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c7">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c7">W1</font> )
 <b>by </b><i><a class="ref" href="pencil_4.html#D6" target="_self" title="PENCIL_4:def.6">Def6</a>, <a class="txt" href="pencil_4.html#E2:35"><i><font color="Green" title="E35">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">W2</font> being    <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="E6:35"/><i><font color="Green" title="E37">A4</font></i>: 
(  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c8">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="vectsp_9.html#K2" title="VECTSP_9:func.2">Subspaces_of</a> <font color="Maroon" title="c8">W2</font> )
 <b>by </b><i><a class="ref" href="pencil_4.html#D6" target="_self" title="PENCIL_4:def.6">Def6</a>, <a class="txt" href="pencil_4.html#E2:35"><i><font color="Green" title="E35">A2</font></i></a></i>;<br/>
<b>assume </b><a NAME="E7:35"/>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">K</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">L</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:35"/><i><font color="Green" title="E38">A5</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">K</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">L</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">K</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">L</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T2" title="PENCIL_1:th.2">PENCIL_1:2</a></i>;<br/>
<a NAME="E10:35"/><i><font color="Green" title="E39">A6</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font> )
 
<b>by </b><i><a class="txt" href="pencil_4.html#E9:35"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">X</font> being  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c7">W1</font><b> such that </b><br/><a NAME="E12:35"/><i><font color="Green" title="E40">A7</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E10:35"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="pencil_4.html#E4:35"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="vectsp_9.html#D3" title="VECTSP_9:def.3">VECTSP_9:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">X'</font> being  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c8">W2</font><b> such that </b><br/><a NAME="E14:35"/><i><font color="Green" title="E41">A8</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">X'</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c12">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E10:35"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="pencil_4.html#E6:35"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="vectsp_9.html#D3" title="VECTSP_9:def.3">VECTSP_9:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">Y</font> being  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c7">W1</font><b> such that </b><br/><a NAME="E16:35"/><i><font color="Green" title="E42">A9</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Y</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c13">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E10:35"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="pencil_4.html#E4:35"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="vectsp_9.html#D3" title="VECTSP_9:def.3">VECTSP_9:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">Y'</font> being  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c8">W2</font><b> such that </b><br/><a NAME="E18:35"/><i><font color="Green" title="E43">A10</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Y'</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c14">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E10:35"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="pencil_4.html#E6:35"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="vectsp_9.html#D3" title="VECTSP_9:def.3">VECTSP_9:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c10">y</font> as  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">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="pencil_4.html#E12:35"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="pencil_4.html#E16:35"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="vectsp_4.html#T34" title="VECTSP_4:th.34">VECTSP_4:34</a></i>;<br/>
<a NAME="E20:35"/>
<font color="Maroon" title="c15">x</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T11" title="VECTSP_5:th.11">VECTSP_5:11</a></i>;<br/>
<a NAME="E21:35"/><b>then </b><i><font color="Green" title="E44">A11</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E12:35"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="vectsp_9.html#T29" title="VECTSP_9:th.29">VECTSP_9:29</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1"/><i><font color="Green" title="E45">A12</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c17">x'</font> = <font color="Maroon" title="c15">x</font> as  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font> <b>by </b><i><a class="ref" href="vectsp_5.html#T11" title="VECTSP_5:th.11">VECTSP_5:11</a></i>;<br/><a NAME="E3:35_1"/>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c17">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E12:35"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="pencil_4.html#E1:35_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="vectsp_9.html#T32" title="VECTSP_9:th.32">VECTSP_9:32</a></i>;<br/><a NAME="E4:35_1"/><b>then </b>
<font color="Maroon" title="c17">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font></span>)</span>
 ;<br/><a NAME="E5:35_1"/><b>then </b>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font>
 ;<br/><a NAME="E6:35_1"/><b>then </b>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T9" title="VECTSP_5:th.9">VECTSP_5:9</a></i>;<br/><a NAME="E7:35_1"/><b>then </b><i><font color="Green" title="E46">A13</font></i>: 
<font color="Maroon" title="c16">y</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T12" title="VECTSP_5:th.12">VECTSP_5:12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">y'</font> = <font color="Maroon" title="c16">y</font> as  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font> <b>by </b><i><a class="ref" href="vectsp_5.html#T11" title="VECTSP_5:th.11">VECTSP_5:11</a></i>;<br/><a NAME="E9:35_1"/>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c18">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E16:35"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="pencil_4.html#E1:35_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="vectsp_9.html#T32" title="VECTSP_9:th.32">VECTSP_9:32</a></i>;<br/><a NAME="E10:35_1"/><b>then </b>
<font color="Maroon" title="c18">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font></span>)</span>
 ;<br/><a NAME="E11:35_1"/><b>then </b>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font>
 ;<br/><a NAME="E12:35_1"/><b>then </b>
<font color="Maroon" title="c15">x</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c16">y</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T12" title="VECTSP_5:th.12">VECTSP_5:12</a></i>;<br/><b>hence </b><a NAME="E13:35_1"/>
contradiction
 <b>by </b><i><a class="ref" href="vectsp_4.html#T33" title="VECTSP_4:th.33">VECTSP_4:33</a>, <a class="txt" href="pencil_4.html#E7:35_1"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="pencil_4.html#E9:35"><i><font color="Green" title="E38">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E23:35"/><b>then </b><i><font color="Green" title="E45">A14</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E21:35"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E24:35"/>
<font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c7">W1</font>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E12:35"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="pencil_4.html#E16:35"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="vectsp_5.html#T40" title="VECTSP_5:th.40">VECTSP_5:40</a></i>;<br/>
<a NAME="E25:35"/><b>then </b>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T29" title="VECTSP_9:th.29">VECTSP_9:29</a>, <a class="txt" href="pencil_4.html#E4:35"><i><font color="Green" title="E36">A3</font></i></a></i>;<br/>
<a NAME="E26:35"/><b>then </b><i><font color="Green" title="E46">A15</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="pencil_4.html#E23:35"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="nat_1.html#T9" title="NAT_1:th.9">NAT_1:9</a></i>;<br/>
<a NAME="E27:35"/><i><font color="Green" title="E47">A16</font></i>: 
<font color="Maroon" title="c11">X</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c13">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E12:35"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="pencil_4.html#E16:35"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="vectsp10.html#T13" title="VECTSP10:th.13">VECTSP10:13</a></i>;<br/>
<a NAME="E28:35"/><i><font color="Green" title="E48">A17</font></i>: 
<font color="Maroon" title="c12">X'</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c14">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</font>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E14:35"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="pencil_4.html#E18:35"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="vectsp10.html#T13" title="VECTSP10:th.13">VECTSP10:13</a></i>;<br/>
<a NAME="E29:35"/>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">X</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c13">Y</font></span>)</span> <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="c7">W1</font>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E27:35"><i><font color="Green" title="E47">A16</font></i></a>, <a class="txt" href="pencil_4.html#E26:35"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="pencil_4.html#E4:35"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="vectsp_9.html#T32" title="VECTSP_9:th.32">VECTSP_9:32</a></i>;<br/>
<a NAME="E30:35"/><b>then </b><i><font color="Green" title="E49">A18</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</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="c7">W1</font>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E27:35"><i><font color="Green" title="E47">A16</font></i></a></i>;<br/>
<a NAME="E31:35"/>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">X'</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c14">Y'</font></span>)</span> <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="c8">W2</font>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E28:35"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="pencil_4.html#E26:35"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="pencil_4.html#E6:35"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="vectsp_9.html#T32" title="VECTSP_9:th.32">VECTSP_9:32</a></i>;<br/>
<a NAME="E32:35"/><b>then </b><i><font color="Green" title="E50">A19</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c16">y</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="c8">W2</font>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E28:35"><i><font color="Green" title="E48">A17</font></i></a></i>;<br/>
<b>thus </b><a NAME="E33:35"/>
<font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E30:35"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="pencil_4.html#E32:35"><i><font color="Green" title="E50">A19</font></i></a>, <a class="ref" href="pencil_4.html#T25" target="_self" title="PENCIL_4:th.25">Th25</a>, <a class="txt" href="pencil_4.html#E4:35"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="pencil_4.html#E6:35"><i><font color="Green" title="E37">A4</font></i></a></i>;<br/>


</div>
