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<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">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#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>;<br/><b>let </b><font color="Maroon" title="c3">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E21">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> &amp; 3 <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#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">k</font>;<br/>
<b>thus </b><a NAME="E2:23"/>
not  <a href="pencil_4.html#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">k</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">B</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#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">k</font></span>)</span>;<br/><a NAME="E1:23_1_1"/>
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#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">k</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:23"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="pencil_4.html#T11" target="_self" title="PENCIL_4:th.11">Th11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">W1</font>, <font color="Maroon" title="c7">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="E3:23_1_1"/><i><font color="Green" title="E22">A2</font></i>: 
( <font color="Maroon" title="c6">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c7">W2</font> &amp; <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c7">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_4.html#K3" title="PENCIL_4:func.3">pencil</a> <font color="Maroon" title="c6">W1</font>,<font color="Maroon" title="c7">W2</font>,<font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="ref" href="pencil_4.html#D4" target="_self" title="PENCIL_4:def.4">Def4</a></i>;<br/><a NAME="E4:23_1_1"/><i><font color="Green" title="E23">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> or ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> ) )
 </span><b>by </b><i><a class="txt" href="pencil_4.html#E1:23"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="pencil_4.html#T1" target="_self" title="PENCIL_4:th.1">Th1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_1"/><b>then </b><i><font color="Green" title="E24">A4</font></i>: 
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c7">W2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:23_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="vectsp_9.html#T32" title="VECTSP_9:th.32">VECTSP_9:32</a></i>;<br/><a NAME="E3:23_1_1_1_1"/><i><font color="Green" title="E25">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_1_1_1"/><i><font color="Green" title="E26">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">W2</font>
 ;<br/><a NAME="E2:23_1_1_1_1_1"/>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c7">W2</font> is    <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="ref" href="pencil_4.html#T5" target="_self" title="PENCIL_4:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_4.html#E2:23_1_1_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="pencil_4.html#E1:23_1_1_1_1_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="vectsp_4.html#T37" title="VECTSP_4:th.37">VECTSP_4:37</a></i>;<br/></div><b>end;</b></div><a NAME="E5:23_1_1_1_1"/><b>then </b>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">W2</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:23_1_1_1_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:23_1_1_1_1"/><i><font color="Green" title="E26">A7</font></i>: 
( <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">W2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">v</font> = <font color="Maroon" title="c8">v</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="pencil_4.html#E7:23_1_1_1_1"><i><font color="Green" title="E26">A7</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c10">X</font> = <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/><a NAME="E9:23_1_1_1_1"/>
not <font color="Maroon" title="c9">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">W2</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E7:23_1_1_1_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E10:23_1_1_1_1"/><b>then </b><i><font color="Green" title="E27">A8</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="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:23_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="pencil_4.html#T14" target="_self" title="PENCIL_4:th.14">Th14</a>, <a class="ref" href="vectsp_4.html#T16" title="VECTSP_4:th.16">VECTSP_4:16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_1_1_2"/>
<font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ;<br/><a NAME="E2:23_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c7">W2</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:23_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="pencil_4.html#T9" target="_self" title="PENCIL_4:th.9">Th9</a></i>;<br/><a NAME="E3:23_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E28">A9</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">W2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/><a NAME="E4:23_1_1_1_1_2"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:23_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T13" title="VECTSP_7:th.13">VECTSP_7:13</a></i>;<br/><a NAME="E6:23_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T6" title="VECTSP_5:th.6">VECTSP_5:6</a></i>;<br/><a NAME="E7:23_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>hence </b><a NAME="E8:23_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_4.html#E7:23_1_1_1_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="pencil_4.html#E3:23_1_1_1_1_2"><i><font color="Green" title="E28">A9</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:23_1_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E10:23_1_1_1_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="vectsp_9.html#D3" title="VECTSP_9:def.3">VECTSP_9:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_1_2"/><i><font color="Green" title="E24">A10</font></i>: 
( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> )
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> &amp;  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:23"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:23_1_1_1_2"/><b>then </b><i><font color="Green" title="E25">A11</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">T</font> being    <a href="vectsp_5.html#M1" title="VECTSP_5:mode.1">Linear_Compl</a> of <font color="Maroon" title="c6">W1</font>;<br/><a NAME="E5:23_1_1_1_2"/><i><font color="Green" title="E26">A12</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/><a NAME="E6:23_1_1_1_2"/><i><font color="Green" title="E27">A13</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c6">W1</font>,<font color="Maroon" title="c8">T</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#T48" title="VECTSP_5:th.48">VECTSP_5:48</a></i>;<br/><a NAME="E7:23_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c8">T</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:23_1_1_1_2"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="vectsp_9.html#T38" title="VECTSP_9:th.38">VECTSP_9:38</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">u</font>, <font color="Maroon" title="c10">v</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c8">T</font><b> such that </b><br/><a NAME="E9:23_1_1_1_2"/><i><font color="Green" title="E28">A14</font></i>: 
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">v</font> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> &amp;  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:23_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="vectsp_9.html#T35" title="VECTSP_9:th.35">VECTSP_9:35</a></i>;<br/><a NAME="E10:23_1_1_1_2"/><i><font color="Green" title="E29">A15</font></i>: 
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c8">T</font> &amp; <font color="Maroon" title="c10">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c8">T</font> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E9:23_1_1_1_2"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="vectsp_7.html#T6" title="VECTSP_7:th.6">VECTSP_7:6</a></i>;<br/><a NAME="E11:23_1_1_1_2"/>
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">T</font> &amp; <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">T</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c11">u1</font> = <font color="Maroon" title="c9">u</font>, <font color="Maroon" title="c12">v1</font> = <font color="Maroon" title="c10">v</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="pencil_4.html#E5:23_1_1_1_2"><i><font color="Green" title="E26">A12</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c13">I</font> being    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c6">W1</font>;<br/><b>reconsider </b><font color="Maroon" title="c14">I1</font> = <font color="Maroon" title="c13">I</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">W1</font> <b>by </b><i><a class="ref" href="vectsp_9.html#T24" title="VECTSP_9:th.24">VECTSP_9:24</a></i>;<br/><a NAME="E15:23_1_1_1_2"/>
2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:23_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="pencil_4.html#E1:23_1_1_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E16:23_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">I1</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#D2" title="VECTSP_9:def.2">VECTSP_9:def 2</a></i>;<br/><a NAME="E17:23_1_1_1_2"/><b>then </b>
not <font color="Maroon" title="c14">I1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="card_4.html#T17" title="CARD_4:th.17">CARD_4:17</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E19:23_1_1_1_2"/><i><font color="Green" title="E30">A16</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">I</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">i</font> = <font color="Maroon" title="c15">i</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c6">W1</font> <b>by </b><i><a class="txt" href="pencil_4.html#E19:23_1_1_1_2"><i><font color="Green" title="E30">A16</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">J</font> = <font color="Maroon" title="c14">I1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="pencil_4.html#E4:23_1_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E22:23_1_1_1_2"/><i><font color="Green" title="E31">A17</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c13">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#D2" title="VECTSP_9:def.2">VECTSP_9:def 2</a></i>;<br/><i><font color="Green" title="E32">A18</font></i>:  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">J</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">I1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="pencil_4.html#E19:23_1_1_1_2"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="euler_1.html#T5" title="EULER_1:th.5">EULER_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="txt" href="pencil_4.html#E22:23_1_1_1_2"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>
;<br/><a NAME="E24:23_1_1_1_2"/><i><font color="Green" title="E33">A19</font></i>: 
<font color="Maroon" title="c17">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">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><br/>

<b>assume </b><a NAME="E1:23_1_1_1_2_2"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">J</font>
 ;<br/>

<a NAME="E2:23_1_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c18">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#T13" title="VECTSP_7:th.13">VECTSP_7:13</a></i>;<br/>
<b>hence </b><a NAME="E3:23_1_1_1_2_2"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E25:23_1_1_1_2"/>
<font color="Maroon" title="c13">I</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/><a NAME="E26:23_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">I</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T2" title="VECTSP_7:th.2">VECTSP_7:2</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">JJ</font> = <font color="Maroon" title="c13">I</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="vectsp_9.html#T15" title="VECTSP_9:th.15">VECTSP_9:15</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">JJJ</font> = <font color="Maroon" title="c18">JJ</font> as  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <b>by </b><i><a class="txt" href="pencil_4.html#E24:23_1_1_1_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="vectsp_9.html#T16" title="VECTSP_9:th.16">VECTSP_9:16</a></i>;<br/><a NAME="E29:23_1_1_1_2"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c19">JJJ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a></i>;<br/><a NAME="E30:23_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">J</font> is    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/><a NAME="E31:23_1_1_1_2"/><b>then </b><i><font color="Green" title="E34">A20</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="pencil_4.html#E23:23_1_1_1_2"><i><font color="Green" title="E32">A18</font></i></a>, <a class="ref" href="vectsp_9.html#D2" title="VECTSP_9:def.2">VECTSP_9:def 2</a></i>;<br/><a NAME="E32:23_1_1_1_2"/><i><font color="Green" title="E35">A21</font></i>: 
<font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E6:23_1_1_1_2"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="vectsp_5.html#D4" title="VECTSP_5:def.4">VECTSP_5:def 4</a></i>;<br/><div><i><font color="Green" title="E36">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_1_2_3"/>
<font color="Maroon" title="c11">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font>
 ;<br/><a NAME="E2:23_1_1_1_2_3"/><b>then </b>
( <font color="Maroon" title="c11">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> &amp; <font color="Maroon" title="c11">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">T</font> )
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E3:23_1_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c11">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E32:23_1_1_1_2"><i><font color="Green" title="E35">A21</font></i></a>, <a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a></i>;<br/><a NAME="E4:23_1_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c11">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E5:23_1_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c11">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></i>;<br/><a NAME="E6:23_1_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c11">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_4.html#E10:23_1_1_1_2"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="vectsp_4.html#T19" title="VECTSP_4:th.19">VECTSP_4:19</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E37">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_1_2_4"/>
<font color="Maroon" title="c12">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font>
 ;<br/><a NAME="E2:23_1_1_1_2_4"/><b>then </b>
( <font color="Maroon" title="c12">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> &amp; <font color="Maroon" title="c12">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">T</font> )
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E3:23_1_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c12">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E32:23_1_1_1_2"><i><font color="Green" title="E35">A21</font></i></a>, <a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a></i>;<br/><a NAME="E4:23_1_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c12">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E5:23_1_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c12">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></i>;<br/><a NAME="E6:23_1_1_1_2_4"/><b>then </b>
<font color="Maroon" title="c12">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_1_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_4.html#E10:23_1_1_1_2"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="vectsp_4.html#T19" title="VECTSP_4:th.19">VECTSP_4:19</a></i>;<br/></div><b>end;</b></div><a NAME="E35:23_1_1_1_2"/><i><font color="Green" title="E38">A24</font></i>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c13">I</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">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/><a NAME="E36:23_1_1_1_2"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c13">I</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T18" title="VECTSP_7:th.18">VECTSP_7:18</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E37:23_1_1_1_2"/><b>then </b><i><font color="Green" title="E39">A25</font></i>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">W1</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a></i>;<br/><a NAME="E38:23_1_1_1_2"/><b>then </b><i><font color="Green" title="E40">A26</font></i>: 
not <font color="Maroon" title="c11">u1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E33:23_1_1_1_2"><i><font color="Green" title="E36">A22</font></i></a>, <a class="ref" href="vectsp_4.html#T16" title="VECTSP_4:th.16">VECTSP_4:16</a></i>;<br/><a NAME="E39:23_1_1_1_2"/><i><font color="Green" title="E41">A27</font></i>: 
not <font color="Maroon" title="c12">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E34:23_1_1_1_2"><i><font color="Green" title="E37">A23</font></i></a>, <a class="txt" href="pencil_4.html#E37:23_1_1_1_2"><i><font color="Green" title="E39">A25</font></i></a>, <a class="ref" href="vectsp_4.html#T16" title="VECTSP_4:th.16">VECTSP_4:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">Y</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="pencil_4.html#E9:23_1_1_1_2"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="vectsp_9.html#T15" title="VECTSP_9:th.15">VECTSP_9:15</a></i>;<br/><a NAME="E41:23_1_1_1_2"/><i><font color="Green" title="E42">A28</font></i>: 
<font color="Maroon" title="c20">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/><a NAME="E42:23_1_1_1_2"/><i><font color="Green" title="E43">A29</font></i>: 
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T50" title="VECTSP_4:th.50">VECTSP_4:50</a></i>;<br/><a NAME="E43:23_1_1_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">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><br/>

<b>assume </b><a NAME="E1:23_1_1_1_2_5"/>
<font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:23_1_1_1_2_5"/><b>then </b>
<font color="Maroon" title="c21">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E3:23_1_1_1_2_5"/><b>then </b>
( <font color="Maroon" title="c21">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font> &amp; <font color="Maroon" title="c21">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a></i>;<br/>
<a NAME="E4:23_1_1_1_2_5"/><b>then </b>
( <font color="Maroon" title="c21">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> &amp; <font color="Maroon" title="c21">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="txt" href="pencil_4.html#E37:23_1_1_1_2"><i><font color="Green" title="E39">A25</font></i></a>, <a class="ref" href="vectsp_4.html#T16" title="VECTSP_4:th.16">VECTSP_4:16</a>, <a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a></i>;<br/>
<a NAME="E5:23_1_1_1_2_5"/><b>then </b>
( <font color="Maroon" title="c21">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> &amp; <font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">T</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U4</a> of <font color="Maroon" title="c8">T</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c8">T</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c8">T</font> #) )
 
<b>by </b><i><a class="txt" href="pencil_4.html#E9:23_1_1_1_2"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E6:23_1_1_1_2_5"/><b>then </b>
( <font color="Maroon" title="c21">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> &amp; <font color="Maroon" title="c21">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">T</font> )
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E7:23_1_1_1_2_5"/><b>then </b>
<font color="Maroon" title="c21">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c8">T</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a></i>;<br/>
<b>hence </b><a NAME="E8:23_1_1_1_2_5"/>
<font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E32:23_1_1_1_2"><i><font color="Green" title="E35">A21</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E44:23_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T35" title="VECTSP_4:th.35">VECTSP_4:35</a></i>;<br/><a NAME="E45:23_1_1_1_2"/><b>then </b><i><font color="Green" title="E44">A30</font></i>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 <b>by </b><i><a class="txt" href="pencil_4.html#E9:23_1_1_1_2"><i><font color="Green" title="E28">A14</font></i></a>, <a class="txt" href="pencil_4.html#E31:23_1_1_1_2"><i><font color="Green" title="E34">A20</font></i></a>, <a class="txt" href="pencil_4.html#E38:23_1_1_1_2"><i><font color="Green" title="E40">A26</font></i></a>, <a class="txt" href="pencil_4.html#E39:23_1_1_1_2"><i><font color="Green" title="E41">A27</font></i></a>, <a class="txt" href="pencil_4.html#E41:23_1_1_1_2"><i><font color="Green" title="E42">A28</font></i></a>, <a class="txt" href="pencil_4.html#E42:23_1_1_1_2"><i><font color="Green" title="E43">A29</font></i></a>, <a class="ref" href="pencil_4.html#T15" target="_self" title="PENCIL_4:th.15">Th15</a>, <a class="ref" href="vectsp_4.html#T33" title="VECTSP_4:th.33">VECTSP_4:33</a></i>;<br/><a NAME="E46:23_1_1_1_2"/><i><font color="Green" title="E45">A31</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_1_2_6"/>
<span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ;<br/><a NAME="E2:23_1_1_1_2_6"/><b>then </b><i><font color="Green" title="E46">A32</font></i>: 
<font color="Maroon" title="c6">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:23_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="pencil_4.html#T9" target="_self" title="PENCIL_4:th.9">Th9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_1_2_6_1"/><i><font color="Green" title="E47">A33</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c21">IV</font> = <font color="Maroon" title="c13">I</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="pencil_4.html#E4:23_1_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:23_1_1_1_2_6_1"/>
<font color="Maroon" title="c21">IV</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">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><br/>

<b>assume </b><a NAME="E1:23_1_1_1_2_6_1_1"/><i><font color="Green" title="E48">A34</font></i>: 
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">IV</font>
 ;<br/>

<a NAME="E2:23_1_1_1_2_6_1_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">I</font>
 
<b>by </b><i><a class="txt" href="pencil_4.html#E19:23_1_1_1_2"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E3:23_1_1_1_2_6_1_1"/><b>then </b><i><font color="Green" title="E49">A35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_1_2_6_1_1_1"/>
( <font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">J</font> or <font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span><b>by </b><i><a class="txt" href="pencil_4.html#E1:23_1_1_1_2_6_1_1"><i><font color="Green" title="E48">A34</font></i></a>, <a class="txt" href="pencil_4.html#E3:23_1_1_1_2_6_1_1"><i><font color="Green" title="E49">A35</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1_2_6_1_1_1_1"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">J</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_2_6_1_1_1_1"/><b>then </b><i><font color="Green" title="E50">A36</font></i>: 
<font color="Maroon" title="c22">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T13" title="VECTSP_7:th.13">VECTSP_7:13</a></i>;<br/><a NAME="E3:23_1_1_1_2_6_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c22">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T17" title="VECTSP_4:th.17">VECTSP_4:17</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c23">o</font> = <font color="Maroon" title="c22">a</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E5:23_1_1_1_2_6_1_1_1_1"/>
<font color="Maroon" title="c23">o</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E2:23_1_1_1_2_6_1_1_1_1"><i><font color="Green" title="E50">A36</font></i></a>, <a class="ref" href="vectsp_5.html#T6" title="VECTSP_5:th.6">VECTSP_5:6</a></i>;<br/><b>hence </b><a NAME="E6:23_1_1_1_2_6_1_1_1_1"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_1_2_6_1_1_1_2"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">i</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_2_6_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_1_2_6_1_1_1_2"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E1:23_1_1_1_2_6_1"><i><font color="Green" title="E47">A33</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:23_1_1_1_2_6_1"/><b>then </b>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c21">IV</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T20" title="VECTSP_9:th.20">VECTSP_9:20</a></i>;<br/><a NAME="E5:23_1_1_1_2_6_1"/><b>then </b>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c13">I</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a></i>;<br/><a NAME="E6:23_1_1_1_2_6_1"/><b>then </b><i><font color="Green" title="E48">A37</font></i>: 
<font color="Maroon" title="c6">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E35:23_1_1_1_2"><i><font color="Green" title="E38">A24</font></i></a>, <a class="ref" href="pencil_4.html#T6" target="_self" title="PENCIL_4:th.6">Th6</a></i>;<br/><a NAME="E7:23_1_1_1_2_6_1"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <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="E8:23_1_1_1_2_6_1"/><b>then </b><i><font color="Green" title="E49">A38</font></i>: 
<font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E6:23_1_1_1_2_6_1"><i><font color="Green" title="E48">A37</font></i></a>, <a class="ref" href="vectsp_5.html#T40" title="VECTSP_5:th.40">VECTSP_5:40</a></i>;<br/><a NAME="E9:23_1_1_1_2_6_1"/><i><font color="Green" title="E50">A39</font></i>: 
 <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T50" title="VECTSP_4:th.50">VECTSP_4:50</a></i>;<br/><a NAME="E10:23_1_1_1_2_6_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_2_6_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">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><br/>

<b>assume </b><a NAME="E1:23_1_1_1_2_6_1_2"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:23_1_1_1_2_6_1_2"/><b>then </b>
<font color="Maroon" title="c22">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E3:23_1_1_1_2_6_1_2"/><b>then </b>
( <font color="Maroon" title="c22">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> &amp; <font color="Maroon" title="c22">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a></i>;<br/>
<a NAME="E4:23_1_1_1_2_6_1_2"/><b>then </b>
( <font color="Maroon" title="c22">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> &amp; <font color="Maroon" title="c22">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a></i>;<br/>
<a NAME="E5:23_1_1_1_2_6_1_2"/><b>then </b>
( <font color="Maroon" title="c22">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> &amp; <font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">VectSpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">T</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U4</a> of <font color="Maroon" title="c8">T</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c8">T</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c8">T</font> #) )
 
<b>by </b><i><a class="txt" href="pencil_4.html#E9:23_1_1_1_2"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E6:23_1_1_1_2_6_1_2"/><b>then </b>
( <font color="Maroon" title="c22">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> &amp; <font color="Maroon" title="c22">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c8">T</font> )
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E7:23_1_1_1_2_6_1_2"/><b>then </b>
<font color="Maroon" title="c22">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c8">T</font>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a></i>;<br/>
<b>hence </b><a NAME="E8:23_1_1_1_2_6_1_2"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_4.html#E32:23_1_1_1_2"><i><font color="Green" title="E35">A21</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:23_1_1_1_2_6_1"/><b>then </b>
<font color="Maroon" title="c6">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T35" title="VECTSP_4:th.35">VECTSP_4:35</a></i>;<br/><a NAME="E12:23_1_1_1_2_6_1"/><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="c6">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">u1</font>,<font color="Maroon" title="c12">v1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="txt" href="pencil_4.html#E9:23_1_1_1_2"><i><font color="Green" title="E28">A14</font></i></a>, <a class="txt" href="pencil_4.html#E33:23_1_1_1_2"><i><font color="Green" title="E36">A22</font></i></a>, <a class="txt" href="pencil_4.html#E34:23_1_1_1_2"><i><font color="Green" title="E37">A23</font></i></a>, <a class="txt" href="pencil_4.html#E41:23_1_1_1_2"><i><font color="Green" title="E42">A28</font></i></a>, <a class="txt" href="pencil_4.html#E9:23_1_1_1_2_6_1"><i><font color="Green" title="E50">A39</font></i></a>, <a class="ref" href="pencil_4.html#T15" target="_self" title="PENCIL_4:th.15">Th15</a>, <a class="ref" href="vectsp_4.html#T33" title="VECTSP_4:th.33">VECTSP_4:33</a></i>;<br/><a NAME="E13:23_1_1_1_2_6_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c6">W1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="pencil_4.html#E45:23_1_1_1_2"><i><font color="Green" title="E44">A30</font></i></a>, <a class="txt" href="pencil_4.html#E8:23_1_1_1_2_6_1"><i><font color="Green" title="E49">A38</font></i></a>, <a class="ref" href="vectsp_9.html#T29" title="VECTSP_9:th.29">VECTSP_9:29</a></i>;<br/><b>hence </b><a NAME="E14:23_1_1_1_2_6_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:23_1_1_1_2_6"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_4.html#E46:23_1_1_1_2"><i><font color="Green" title="E45">A31</font></i></a>, <a class="txt" href="pencil_4.html#E2:23_1_1_1_2_6"><i><font color="Green" title="E46">A32</font></i></a>, <a class="ref" href="vectsp_4.html#T16" title="VECTSP_4:th.16">VECTSP_4:16</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E48:23_1_1_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E3:23_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="pencil_4.html#E45:23_1_1_1_2"><i><font color="Green" title="E44">A30</font></i></a>, <a class="ref" href="vectsp_9.html#D3" title="VECTSP_9:def.3">VECTSP_9:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E2:23_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_4.html#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">k</font></span>)</span> is  not   <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#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">k</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E3:23_1"/>
not  <a href="pencil_4.html#K5" title="PENCIL_4:func.5">PencilSpace</a> <font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">k</font> is <a href="pencil_1.html#V4" title="PENCIL_1:attr.4">degenerated</a>
 <b>by </b><i><a class="ref" href="pencil_1.html#D5" title="PENCIL_1:def.5">PENCIL_1:def 5</a></i>;<br/>


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


</div>
