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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <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="c1">V</font>  st <font color="Olive" title="b1">A</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  &amp;  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Olive" title="b2">B</font> holds <br/> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">A</font> <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="Olive" title="b2">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <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="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  &amp;  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">B</font> implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font> <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="c3">B</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:104"/><span class="lab"><font color="Green" title="E77">A1</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a> 
 <span class="kw">and </span><br/><a NAME="E2:104"/><span class="lab"><font color="Green" title="E78">A2</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font> <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="c3">B</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:104_1"/>
( <font color="Maroon" title="c2">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c2">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1"><span class="kw">suppose </span></a><a NAME="E1:104_1_1"/>
<font color="Maroon" title="c2">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font> <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="c3">B</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:104_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font> <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="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_2"><span class="kw">suppose </span></a><a NAME="E1:104_1_2"/>
not <font color="Maroon" title="c2">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font> <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="c3">B</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:104_1_2"/><span class="lab"><font color="Green" title="E79">A3</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104_1_2"><span class="lab"><font color="Green" title="E79">A3</font></span></a></span>;<br/><a NAME="E5:104_1_2"/><span class="lab"><font color="Green" title="E80">A4</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="rlaffin1.html#E61"><span class="lab"><font color="Green" title="E50">Lm7</font></span></a></span>;<br/><a NAME="E6:104_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A5</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104_1_2"><span class="lab"><font color="Green" title="E79">A3</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">LA</font> =  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</font></span>)</span>;<br/><a NAME="E7:104_1_2"/><span class="lab"><font color="Green" title="E82">A6</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T7" target="_self" title="RLAFFIN1:th.7">Th7</a></span>;<br/><a NAME="E8:104_1_2"/>
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">Ib</font> being   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E10:104_1_2"/>
 <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ib</font>
 <span class="kw">and </span><br/><a NAME="E11:104_1_2"/><span class="lab"><font color="Green" title="E83">A7</font></span>: 
<font color="Maroon" title="c7">Ib</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E12:104_1_2"/><span class="lab"><font color="Green" title="E84">A8</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c7">Ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T60" target="_self" title="RLAFFIN1:th.60">Th60</a></span>;<br/><a NAME="E13:104_1_2"/>
not <font color="Maroon" title="c7">Ib</font> 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="#E2:104"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="txt" href="#E3:104_1_2"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="txt" href="#E12:104_1_2"><span class="lab"><font color="Green" title="E84">A8</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E15:104_1_2"/><span class="lab"><font color="Green" title="E85">A9</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Ib</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:104_1_2"><span class="lab"><font color="Green" title="E85">A9</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">LI</font> =  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font></span>)</span>;<br/><a NAME="E17:104_1_2"/><span class="lab"><font color="Green" title="E86">A10</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">Ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T7" target="_self" title="RLAFFIN1:th.7">Th7</a></span>;<br/><a NAME="E18:104_1_2"/>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</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="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font> or <font color="Maroon" title="c11">x</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="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:104_1_2_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</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="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:104_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_3.html#T18" title="RLVECT_3:th.18">RLVECT_3:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:104_1_2_1"/>
<font color="Maroon" title="c11">x</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="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font></span>)</span></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="Red">::  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="c11">qI</font> = <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</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 <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:104_1_2"><span class="lab"><font color="Green" title="E83">A7</font></span></a>, <a class="txt" href="#E17:104_1_2"><span class="lab"><font color="Green" title="E86">A10</font></span></a></span>;<br/><a NAME="E20:104_1_2"/>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">q</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="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>;<br/><a NAME="E21:104_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A11</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">qI</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:104_1_2"><span class="lab"><font color="Green" title="E85">A9</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A12</font></span>:  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font></span>)</span> = 
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></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="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T140" title="ZFMISC_1:th.140">ZFMISC_1:140</a></span>
<br/>.= 
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font></span>)</span> <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"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="rlaffin1.html#E72"><span class="lab"><font color="Green" title="E58">Lm9</font></span></a></span>
<br/>.= 
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">qI</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"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_5.html#T21" title="RLVECT_5:th.21">RLVECT_5:21</a></span>
;<br/><a NAME="E23:104_1_2"/><span class="lab"><font color="Green" title="E89">A13</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:104_1_2"><span class="lab"><font color="Green" title="E85">A9</font></span></a>, <a class="ref" href="rlaffin1.html#T41" target="_self" title="RLAFFIN1:th.41">Th41</a></span>;<br/><a NAME="E24:104_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">qI</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"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_5.html#T16" title="RLVECT_5:th.16">RLVECT_5:16</a></span>;<br/><a NAME="E25:104_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">qI</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"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is    <a href="rlvect_3.html#M1" title="RLVECT_3:mode.1">Basis</a> of  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:104_1_2"><span class="lab"><font color="Green" title="E88">A12</font></span></a>, <a class="ref" href="rlvect_3.html#D3" title="RLVECT_3:def.3">RLVECT_3:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">LI</font> =  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c7">Ib</font></span>)</span> as   <a href="rlvect_5.html#V1" title="RLVECT_5:attr.1">finite-dimensional</a>   <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_5.html#D1" title="RLVECT_5:def.1">RLVECT_5:def 1</a></span>;<br/><a NAME="E27:104_1_2"/><span class="lab"><font color="Green" title="E90">A14</font></span>: 
<font color="Maroon" title="c7">Ib</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c7">Ib</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="rlaffin1.html#E61"><span class="lab"><font color="Green" title="E50">Lm7</font></span></a></span>;<br/><a NAME="E28:104_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A15</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c7">Ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c12">LI</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:104_1_2"><span class="lab"><font color="Green" title="E85">A9</font></span></a>, <a class="ref" href="rlaffin1.html#T57" target="_self" title="RLAFFIN1:th.57">Th57</a></span>;<br/><a NAME="E29:104_1_2"/><span class="lab"><font color="Green" title="E92">A16</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104_1_2"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="txt" href="#E5:104_1_2"><span class="lab"><font color="Green" title="E80">A4</font></span></a>, <a class="ref" href="rlaffin1.html#T57" target="_self" title="RLAFFIN1:th.57">Th57</a></span>;<br/><a NAME="E30:104_1_2"/>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</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="c12">LI</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c13">x</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="c12">LI</font> )</span><br/>

<a NAME="E1:104_1_2_3"/><span class="lab"><font color="Green" title="E93">A17</font></span>: 
2 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
;<br/>
<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T29" title="RLVECT_1:th.29">RLVECT_1:29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D14" title="RLVECT_1:def.14">RLVECT_1:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T42" title="RLVECT_1:th.42">RLVECT_1:42</a></span>
;<br/>
<a NAME="E3:104_1_2_3"/><span class="kw">then </span>
( <font color="Maroon" title="c9">q</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c12">LI</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="rlsub_1.html#K3" title="RLSUB_1:func.3">+</a> <font color="Maroon" title="c12">LI</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">q</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c12">LI</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="txt" href="#E12:104_1_2"><span class="lab"><font color="Green" title="E84">A8</font></span></a>, <a class="txt" href="#E6:104_1_2"><span class="lab"><font color="Green" title="E81">A5</font></span></a>, <a class="txt" href="#E15:104_1_2"><span class="lab"><font color="Green" title="E85">A9</font></span></a>, <a class="txt" href="#E27:104_1_2"><span class="lab"><font color="Green" title="E90">A14</font></span></a>, <a class="txt" href="#E28:104_1_2"><span class="lab"><font color="Green" title="E91">A15</font></span></a>, <a class="txt" href="#E1:104_1_2_3"><span class="lab"><font color="Green" title="E93">A17</font></span></a>, <a class="ref" href="rlaffin1.html#T78" target="_self" title="RLAFFIN1:th.78">Th78</a>, <a class="ref" href="rusub_4.html#T32" title="RUSUB_4:th.32">RUSUB_4:32</a></span>;<br/>
<a NAME="E4:104_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A18</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">LI</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#T77" title="RLSUB_1:th.77">RLSUB_1:77</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:104_1_2_3"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</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="c12">LI</font></span><br/>

<a NAME="E6:104_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A19</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_3.html#T18" title="RLVECT_3:th.18">RLVECT_3:18</a></span>;<br/>
<a NAME="E7:104_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlsub_1.html#T17" title="RLSUB_1:th.17">RLSUB_1:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">w</font> = <font color="Maroon" title="c13">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <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>;<br/>
<a NAME="E9:104_1_2_3"/>
<font color="Maroon" title="c14">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:104_1_2_3"><span class="lab"><font color="Green" title="E95">A19</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E10:104_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:104_1_2"><span class="lab"><font color="Green" title="E92">A16</font></span></a></span>;<br/>
<a NAME="E11:104_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">q</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c12">LI</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="txt" href="#E12:104_1_2"><span class="lab"><font color="Green" title="E84">A8</font></span></a>, <a class="txt" href="#E28:104_1_2"><span class="lab"><font color="Green" title="E91">A15</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E13:104_1_2_3"/><span class="lab"><font color="Green" title="E96">A20</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c14">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">u</font>
 <span class="kw">and </span><br/><a NAME="E14:104_1_2_3"/><span class="lab"><font color="Green" title="E97">A21</font></span>: 
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_4.html#K3" title="RUSUB_4:func.3">Up</a> <font color="Maroon" title="c12">LI</font>
 ;<br/>
<span class="lab"><font color="Green" title="E98">A22</font></span>: <font color="Maroon" title="c14">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:104_1_2_3"><span class="lab"><font color="Green" title="E96">A20</font></span></a>, <a class="ref" href="rlvect_4.html#T1" title="RLVECT_4:th.1">RLVECT_4:1</a></span>
<br/>.= 
<font color="Maroon" title="c9">q</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T42" title="RLVECT_1:th.42">RLVECT_1:42</a></span>
<br/>.= 
<font color="Maroon" title="c15">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T43" title="RLVECT_1:th.43">RLVECT_1:43</a></span>
<br/>.= 
<font color="Maroon" title="c15">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D14" title="RLVECT_1:def.14">RLVECT_1:def 14</a></span>
<br/>.= 
<font color="Maroon" title="c15">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T47" title="RLVECT_1:th.47">RLVECT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c15">u</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D14" title="RLVECT_1:def.14">RLVECT_1:def 14</a></span>
;<br/>
<a NAME="E16:104_1_2_3"/>
<font color="Maroon" title="c15">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">LI</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:104_1_2_3"><span class="lab"><font color="Green" title="E97">A21</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E17:104_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">LI</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:104_1_2_3"><span class="lab"><font color="Green" title="E94">A18</font></span></a>, <a class="txt" href="#E15:104_1_2_3"><span class="lab"><font color="Green" title="E98">A22</font></span></a>, <a class="ref" href="rlsub_1.html#T28" title="RLSUB_1:th.28">RLSUB_1:28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:104_1_2_3"/>
<font color="Maroon" title="c13">x</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="c12">LI</font>
 <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="Red">::  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="c13">LA</font> =  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</font></span>)</span> as    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c12">LI</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_5.html#T20" title="RLVECT_5:th.20">RLVECT_5:20</a></span>;<br/><a NAME="E32:104_1_2"/>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</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="c13">LA</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c14">x</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="c13">LA</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:104_1_2_4"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</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="c13">LA</font></span><br/>

<a NAME="E2:104_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c14">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">LA</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_3.html#T18" title="RLVECT_3:th.18">RLVECT_3:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:104_1_2_4"/>
<font color="Maroon" title="c14">x</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="c13">LA</font>
 <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="Red">::  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="c14">pA</font> = <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</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="c13">LA</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:104_1_2"><span class="lab"><font color="Green" title="E82">A6</font></span></a></span>;<br/><a NAME="E34:104_1_2"/>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">p</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="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>;<br/><a NAME="E35:104_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A23</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">pA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104_1_2"><span class="lab"><font color="Green" title="E79">A3</font></span></a></span>;<br/><a NAME="E36:104_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A24</font></span>: 
<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="c1">V</font></span>)</span></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="c14">pA</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><a NAME="E37:104_1_2"/><span class="lab"><font color="Green" title="E95">A25</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</font></span>)</span> <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"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E3:104_1_2"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="ref" href="rlaffin1.html#T41" target="_self" title="RLAFFIN1:th.41">Th41</a></span>;<br/><a NAME="E38:104_1_2"/><span class="lab"><font color="Green" title="E96">A26</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</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="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></span>;<br/><a NAME="E39:104_1_2"/><span class="lab"><font color="Green" title="E97">A27</font></span>: 
<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="c1">V</font></span>)</span></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="c11">qI</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:104_1_2"><span class="lab"><font color="Green" title="E87">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><span class="lab"><font color="Green" title="E98">A28</font></span>:  <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c12">LI</font> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">qI</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"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:104_1_2"><span class="lab"><font color="Green" title="E89">A13</font></span></a>, <a class="txt" href="#E22:104_1_2"><span class="lab"><font color="Green" title="E88">A12</font></span></a>, <a class="ref" href="rlvect_5.html#T16" title="RLVECT_5:th.16">RLVECT_5:16</a>, <a class="ref" href="rlvect_5.html#T30" title="RLVECT_5:th.30">RLVECT_5:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">qI</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:104_1_2"><span class="lab"><font color="Green" title="E97">A27</font></span></a>, <a class="txt" href="#E38:104_1_2"><span class="lab"><font color="Green" title="E96">A26</font></span></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></span>
;<br/> <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</font></span>)</span> = 
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></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="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:104_1_2"><span class="lab"><font color="Green" title="E93">A23</font></span></a>, <a class="ref" href="zfmisc_1.html#T140" title="ZFMISC_1:th.140">ZFMISC_1:140</a></span>
<br/>.= 
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c2">A</font></span>)</span> <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"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="rlaffin1.html#E72"><span class="lab"><font color="Green" title="E58">Lm9</font></span></a></span>
<br/>.= 
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">pA</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"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_5.html#T21" title="RLVECT_5:th.21">RLVECT_5:21</a></span>
;<br/><span class="kw">then </span> <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <font color="Maroon" title="c13">LA</font> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">pA</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"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:104_1_2"><span class="lab"><font color="Green" title="E95">A25</font></span></a>, <a class="ref" href="rlvect_5.html#T16" title="RLVECT_5:th.16">RLVECT_5:16</a>, <a class="ref" href="rlvect_5.html#T30" title="RLVECT_5:th.30">RLVECT_5:30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:104_1_2"><span class="lab"><font color="Green" title="E82">A6</font></span></a>, <a class="txt" href="#E38:104_1_2"><span class="lab"><font color="Green" title="E96">A26</font></span></a>, <a class="txt" href="#E36:104_1_2"><span class="lab"><font color="Green" title="E94">A24</font></span></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></span>
;<br/><a NAME="E43:104_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">qI</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:104_1_2"><span class="lab"><font color="Green" title="E98">A28</font></span></a>, <a class="ref" href="rlvect_5.html#T29" title="RLVECT_5:th.29">RLVECT_5:29</a></span>;<br/><a NAME="E44:104_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A29</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font> <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="c11">qI</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E45:104_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">qI</font> <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="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:104_1_2"><span class="lab"><font color="Green" title="E83">A7</font></span></a>, <a class="txt" href="#E17:104_1_2"><span class="lab"><font color="Green" title="E86">A10</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E46:104_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font> <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="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:104_1_2"><span class="lab"><font color="Green" title="E99">A29</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
