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

<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="matrlin.html#V1" title="MATRLIN:attr.1">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/>  for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b3">n</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> <font color="Olive" title="b1">V</font> holds <br/> <a href="pencil_4.html#K7" title="PENCIL_4:func.7">GrassmannSpace</a> (<font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">n</font>) is  <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</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> <font color="Maroon" title="c2">V</font> holds <br/> <a href="pencil_4.html#K7" title="PENCIL_4:func.7">GrassmannSpace</a> (<font color="Maroon" title="c2">V</font>,<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font>) is  <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</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> <font color="Maroon" title="c2">V</font> implies  <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>,<font color="Maroon" title="c4">n</font>) is  <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">and </span><br/><a NAME="E2:31"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">and </span><br/><a NAME="E3:31"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c4">n</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> <font color="Maroon" title="c2">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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>,<font color="Maroon" title="c4">n</font>) is  <a href="pencil_1.html#V5" title="PENCIL_1:attr.5">with_non_trivial_blocks</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">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>,<font color="Maroon" title="c4">n</font>);<br/>
<span class="kw">let </span><font color="Maroon" title="c6">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#K7" title="PENCIL_4:func.7">GrassmannSpace</a> (<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pencil_1.html#D6" title="PENCIL_1:def.6">PENCIL_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> 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> <font color="Maroon" title="c6">B</font></span><br/>

<a NAME="E4:31"/>
 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>,<font color="Maroon" title="c4">n</font>)</span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="pencil_4.html#T22" target="_self" title="PENCIL_4:th.22">Th24</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">W</font> being    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font><span class="kw"> such that </span><br/><a NAME="E6:31"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">and </span><br/><a NAME="E7:31"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c6">B</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">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_4.html#D6" target="_self" title="PENCIL_4:def.6">Def6</a></span>;<br/>
<a NAME="E8:31"/>
<font color="Maroon" title="c3">m</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> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">U</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c7">W</font><span class="kw"> such that </span><br/><a NAME="E10:31"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c8">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="vectsp_9.html#T35" title="VECTSP_9:th.35">VECTSP_9:35</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">I</font> =  the    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c8">U</font>;<br/>
<a NAME="E11:31"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a>  the    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c8">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="vectsp_9.html#D1" title="VECTSP_9:def.1">VECTSP_9:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">I9</font> =  the    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c8">U</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="c8">U</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T20" title="VECTSP_9:th.20">VECTSP_9:20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">m</font> = <font color="Maroon" title="c3">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E14:31"/>
1 <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> <font color="Maroon" title="c11">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E15:31"/><span class="kw">then </span>
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>  the    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c8">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:31"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:31"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c8">U</font>
 <span class="kw">and </span><br/><a NAME="E18:31"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c8">U</font>
 <span class="kw">and </span><br/><a NAME="E19:31"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T2" title="PENCIL_1:th.2">PENCIL_1:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">I1</font> = <font color="Maroon" title="c10">I9</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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="c8">U</font> ;<br/>
<a NAME="E21:31"/>
<font color="Maroon" title="c14">I1</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="c14">I1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">I1</font> or <font color="Maroon" title="c15">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="c14">I1</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_1"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">I1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">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="c14">I1</font></span>)</span></span><br/>

<a NAME="E2:31_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">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="c14">I1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#T8" title="VECTSP_7:th.8">VECTSP_7:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:31_1"/>
<font color="Maroon" title="c15">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="c14">I1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">I19</font> = <font color="Maroon" title="c14">I1</font> as   <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="c14">I1</font></span>)</span> ;<br/>
<a NAME="E23:31"/><span class="lab"><font color="Green" title="E39">A11</font></span>: 
 the    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c8">U</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></span>;<br/>
<a NAME="E24:31"/><span class="kw">then </span>
<font color="Maroon" title="c14">I1</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#T1" title="VECTSP_7:th.1">VECTSP_7:1</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">II1</font> = <font color="Maroon" title="c15">I19</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="c14">I1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T12" title="VECTSP_9:th.12">VECTSP_9:12</a></span>;<br/>
<a NAME="E26:31"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c16">II1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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 <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c14">I1</font></span>)</span>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U5</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="c14">I1</font></span>)</span>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</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="c14">I1</font></span>)</span>, the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</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="c14">I1</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T17" title="VECTSP_9:th.17">VECTSP_9:17</a></span>;<br/>
<a NAME="E27:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A12</font></span>: 
<font color="Maroon" title="c14">I1</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="c14">I1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">I2</font> = <font color="Maroon" title="c10">I9</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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="c8">U</font> ;<br/>
<a NAME="E29:31"/>
<font color="Maroon" title="c17">I2</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">I2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">I2</font> or <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">I2</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">I2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">I2</font></span>)</span></span><br/>

<a NAME="E2:31_2"/><span class="kw">then </span>
<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">I2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#T8" title="VECTSP_7:th.8">VECTSP_7:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:31_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">I2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">I29</font> = <font color="Maroon" title="c17">I2</font> as   <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">I2</font></span>)</span> ;<br/>
<a NAME="E31:31"/>
<font color="Maroon" title="c17">I2</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:31"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="ref" href="vectsp_7.html#T1" title="VECTSP_7:th.1">VECTSP_7:1</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">II2</font> = <font color="Maroon" title="c18">I29</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">I2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T12" title="VECTSP_9:th.12">VECTSP_9:12</a></span>;<br/>
<a NAME="E33:31"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c19">II2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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 <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">I2</font></span>)</span>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">U5</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">I2</font></span>)</span>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</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">I2</font></span>)</span>, the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</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">I2</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T17" title="VECTSP_9:th.17">VECTSP_9:17</a></span>;<br/>
<a NAME="E34:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A13</font></span>: 
<font color="Maroon" title="c17">I2</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">I2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></span>;<br/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">I2</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">I9</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="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:31"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="euler_1.html#T4" title="EULER_1:th.4">EULER_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:31"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="card_1.html#T30" title="CARD_1:th.30">CARD_1:30</a></span>
;<br/>
<a NAME="E36:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A14</font></span>: 
 <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">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:31"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="ref" href="vectsp_9.html#D1" title="VECTSP_9:def.1">VECTSP_9:def 1</a></span>;<br/>
<a NAME="E37:31"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">I2</font> is   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c7">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T26" title="VECTSP_4:th.26">VECTSP_4:26</a></span>;<br/>
<a NAME="E38:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A15</font></span>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c17">I2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E36:31"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="ref" href="vectsp_9.html#D2" title="VECTSP_9:def.2">VECTSP_9:def 2</a></span>;<br/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">I1</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">I9</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="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:31"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="euler_1.html#T4" title="EULER_1:th.4">EULER_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:31"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="card_1.html#T30" title="CARD_1:th.30">CARD_1:30</a></span>
;<br/>
<a NAME="E40:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A16</font></span>: 
 <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="c14">I1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:31"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="vectsp_9.html#D1" title="VECTSP_9:def.1">VECTSP_9:def 1</a></span>;<br/>
<a NAME="E41:31"/>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c14">I1</font> is   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c7">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T26" title="VECTSP_4:th.26">VECTSP_4:26</a></span>;<br/>
<a NAME="E42:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A17</font></span>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c14">I1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E40:31"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="vectsp_9.html#D2" title="VECTSP_9:def.2">VECTSP_9:def 2</a></span>;<br/>
<a NAME="E43:31"/>
 not <font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:31"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E44:31"/><span class="kw">then </span>
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">I1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:31"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E45:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A18</font></span>: 
<font color="Maroon" title="c13">j</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="c14">I1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#T8" title="VECTSP_7:th.8">VECTSP_7:8</a></span>;<br/>
<a NAME="E46:31"/>
 not <font color="Maroon" title="c13">j</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">I2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:31"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="txt" href="#E18:31"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="vectsp_9.html#T14" title="VECTSP_9:th.14">VECTSP_9:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E47:31"/>
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> <font color="Maroon" title="c6">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:31"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="txt" href="#E38:31"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E45:31"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="ref" href="pencil_1.html#T2" title="PENCIL_1:th.2">PENCIL_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
