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<span class="kw">let </span><font color="Maroon" title="c1">K</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">V1</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">K</font><br/>  for <font color="Olive" title="b2">W1</font>, <font color="Olive" title="b3">W2</font> being    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V1</font>  st <font color="Olive" title="b2">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Olive" title="b3">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Olive" title="b1">V1</font> holds <br/> for <font color="Olive" title="b4">b1</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b2">W1</font><br/>  for <font color="Olive" title="b5">b2</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b3">W2</font><br/>  for <font color="Olive" title="b6">b</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b2">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Olive" title="b3">W2</font>  st <font color="Olive" title="b6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b5">b2</font> holds <br/> for <font color="Olive" title="b7">v</font>, <font color="Olive" title="b8">v1</font>, <font color="Olive" title="b9">v2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Olive" title="b3">W2</font></span>)</span><br/>  for <font color="Olive" title="b10">w1</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b2">W1</font><br/>  for <font color="Olive" title="b11">w2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b3">W2</font>  st <font color="Olive" title="b7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b9">v2</font> &amp; <font color="Olive" title="b8">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b10">w1</font> &amp; <font color="Olive" title="b9">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b11">w2</font> holds <br/><font color="Olive" title="b7">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b10">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b4">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b11">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b5">b2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V1</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">K</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">W1</font>, <font color="Olive" title="b2">W2</font> being    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V1</font>  st <font color="Olive" title="b1">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Olive" title="b2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V1</font> holds <br/> for <font color="Olive" title="b3">b1</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b1">W1</font><br/>  for <font color="Olive" title="b4">b2</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b2">W2</font><br/>  for <font color="Olive" title="b5">b</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b1">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Olive" title="b2">W2</font>  st <font color="Olive" title="b5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b4">b2</font> holds <br/> for <font color="Olive" title="b6">v</font>, <font color="Olive" title="b7">v1</font>, <font color="Olive" title="b8">v2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Olive" title="b2">W2</font></span>)</span><br/>  for <font color="Olive" title="b9">w1</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b1">W1</font><br/>  for <font color="Olive" title="b10">w2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b2">W2</font>  st <font color="Olive" title="b6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b8">v2</font> &amp; <font color="Olive" title="b7">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">w1</font> &amp; <font color="Olive" title="b8">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b10">w2</font> holds <br/><font color="Olive" title="b6">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b9">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b3">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b10">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b4">b2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V1</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="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V1</font> implies  for <font color="Olive" title="b1">b1</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">W1</font><br/>  for <font color="Olive" title="b2">b2</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">W2</font><br/>  for <font color="Olive" title="b3">b</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font>  st <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">b2</font> holds <br/> for <font color="Olive" title="b4">v</font>, <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><br/>  for <font color="Olive" title="b7">w1</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W1</font><br/>  for <font color="Olive" title="b8">w2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">W2</font>  st <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b6">v2</font> &amp; <font color="Olive" title="b5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">w1</font> &amp; <font color="Olive" title="b6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">w2</font> holds <br/><font color="Olive" title="b4">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b1">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b8">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b2">b2</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:28"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V1</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">b1</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">W1</font><br/>  for <font color="Olive" title="b2">b2</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">W2</font><br/>  for <font color="Olive" title="b3">b</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font>  st <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">b2</font> holds <br/> for <font color="Olive" title="b4">v</font>, <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><br/>  for <font color="Olive" title="b7">w1</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W1</font><br/>  for <font color="Olive" title="b8">w2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">W2</font>  st <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b6">v2</font> &amp; <font color="Olive" title="b5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">w1</font> &amp; <font color="Olive" title="b6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">w2</font> holds <br/><font color="Olive" title="b4">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b1">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b8">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b2">b2</font></span>)</span></span><br/>

<a NAME="E2:28"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></span>;<br/>
<a NAME="E3:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</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#T30" title="CARD_1:th.30">CARD_1:30</a></span>;<br/>
<span class="lab"><font color="Green" title="E18">A3</font></span>: <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c4">W2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T32" title="VECTSP_9:th.32">VECTSP_9:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="ranknull.html#T5" title="RANKNULL:th.5">RANKNULL:5</a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">b1</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">W1</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">b2</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">W2</font><br/>  for <font color="Olive" title="b2">b</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font>  st <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">b2</font> holds <br/> for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">v1</font>, <font color="Olive" title="b5">v2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><br/>  for <font color="Olive" title="b6">w1</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W1</font><br/>  for <font color="Olive" title="b7">w2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">W2</font>  st <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b5">v2</font> &amp; <font color="Olive" title="b4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">w1</font> &amp; <font color="Olive" title="b5">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">w2</font> holds <br/><font color="Olive" title="b3">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b1">b2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">b2</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">W2</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">b</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">b2</font> holds <br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><br/>  for <font color="Olive" title="b5">w1</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W1</font><br/>  for <font color="Olive" title="b6">w2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">W2</font>  st <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">v2</font> &amp; <font color="Olive" title="b3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">w1</font> &amp; <font color="Olive" title="b4">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">w2</font> holds <br/><font color="Olive" title="b2">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">b</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</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="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">b2</font> implies  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><br/>  for <font color="Olive" title="b4">w1</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W1</font><br/>  for <font color="Olive" title="b5">w2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">W2</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w1</font> &amp; <font color="Olive" title="b3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">w2</font> holds <br/><font color="Olive" title="b1">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E5:28"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">b2</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">v</font>, <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span><br/>  for <font color="Olive" title="b4">w1</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W1</font><br/>  for <font color="Olive" title="b5">w2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">W2</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w1</font> &amp; <font color="Olive" title="b3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">w2</font> holds <br/><font color="Olive" title="b1">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">R</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">b</font> as    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D2" title="MATRLIN:def.2">MATRLIN:def 2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">v</font>, <font color="Maroon" title="c10">v1</font>, <font color="Maroon" title="c11">v2</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>; <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">w1</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W1</font><br/>  for <font color="Olive" title="b2">w2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">W2</font>  st <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">v2</font> &amp; <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w2</font> holds <br/><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c12">w1</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">W1</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">w2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">W2</font>  st <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">v2</font> &amp; <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w2</font> holds <br/><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c13">w2</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">W2</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="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">v2</font> &amp; <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">w2</font> implies <font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E7:28"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c11">v2</font> &amp; <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">w2</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="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c14">wb2</font> = <font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">L2</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c4">W2</font><span class="kw"> such that </span><br/><a NAME="E9:28"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c13">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c15">L2</font>
 <span class="kw">and </span><br/><a NAME="E10:28"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c15">L2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">b2</font>
 <span class="kw">and </span><br/><a NAME="E11:28"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
 for <font color="Olive" title="b1">k</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">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D7" title="MATRLIN:def.7">MATRLIN:def 7</a></span>;<br/>
<a NAME="E12:28"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
<font color="Maroon" title="c4">W2</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">K2</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font><span class="kw"> such that </span><br/><a NAME="E14:28"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c15">L2</font>
 <span class="kw">and </span><br/><a NAME="E15:28"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c16">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c15">L2</font>
 <span class="kw">and </span><br/><a NAME="E16:28"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<font color="Maroon" title="c16">K2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="matrlin2.html#E25"><span class="lab"><font color="Green" title="E14">Lm4</font></span></a></span>;<br/>
<a NAME="E17:28"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">b2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="finseq_1.html#T30" title="FINSEQ_1:th.30">FINSEQ_1:30</a></span>;<br/>
<a NAME="E18:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A13</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">K2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">wb1</font> = <font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">vb</font> = <font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">L1</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c3">W1</font><span class="kw"> such that </span><br/><a NAME="E20:28"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon" title="c12">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c19">L1</font>
 <span class="kw">and </span><br/><a NAME="E21:28"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c19">L1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">b1</font>
 <span class="kw">and </span><br/><a NAME="E22:28"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
 for <font color="Olive" title="b1">k</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">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D7" title="MATRLIN:def.7">MATRLIN:def 7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">L</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font><span class="kw"> such that </span><br/><a NAME="E24:28"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c20">L</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c20">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">b</font> )
 <span class="kw">and </span><br/><a NAME="E25:28"/><span class="lab"><font color="Green" title="E33">A18</font></span>: 
 for <font color="Olive" title="b1">k</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">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D7" title="MATRLIN:def.7">MATRLIN:def 7</a></span>;<br/>
<a NAME="E26:28"/><span class="lab"><font color="Green" title="E34">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D7" title="MATRLIN:def.7">MATRLIN:def 7</a></span>;<br/>
<a NAME="E27:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A20</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E28:28"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D7" title="MATRLIN:def.7">MATRLIN:def 7</a></span>;<br/>
<a NAME="E29:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A22</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E30:28"/><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<font color="Maroon" title="c8">R</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="E31:28"/><span class="lab"><font color="Green" title="E39">A24</font></span>: 
<font color="Maroon" title="c3">W1</font> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T7" title="VECTSP_5:th.7">VECTSP_5:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">K1</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font><span class="kw"> such that </span><br/><a NAME="E33:28"/><span class="lab"><font color="Green" title="E40">A25</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c21">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c19">L1</font>
 <span class="kw">and </span><br/><a NAME="E34:28"/><span class="lab"><font color="Green" title="E41">A26</font></span>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c21">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c19">L1</font>
 <span class="kw">and </span><br/><a NAME="E35:28"/><span class="lab"><font color="Green" title="E42">A27</font></span>: 
<font color="Maroon" title="c21">K1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="matrlin2.html#E25"><span class="lab"><font color="Green" title="E14">Lm4</font></span></a></span>;<br/>
<a NAME="E36:28"/><span class="lab"><font color="Green" title="E43">A28</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D7" title="MATRLIN:def.7">MATRLIN:def 7</a></span>;<br/>
<a NAME="E37:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A29</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E38:28"/><span class="lab"><font color="Green" title="E45">A30</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E39:28"/><span class="lab"><font color="Green" title="E46">A31</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">W1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c4">W2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T21" target="_self" title="MATRLIN2:th.21">Th21</a></span>;<br/>
<a NAME="E40:28"/><span class="lab"><font color="Green" title="E47">A32</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T21" target="_self" title="MATRLIN2:th.21">Th21</a></span>;<br/>
<a NAME="E41:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A33</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:28"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="txt" href="#E28:28"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E39:28"><span class="lab"><font color="Green" title="E46">A31</font></span></a>, <a class="txt" href="#E38:28"><span class="lab"><font color="Green" title="E45">A30</font></span></a>, <a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E42:28"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="finseq_1.html#T29" title="FINSEQ_1:th.29">FINSEQ_1:29</a></span>;<br/>
<a NAME="E43:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A34</font></span>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c21">K1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:28"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E33:28"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E44:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A35</font></span>: 
<font color="Maroon" title="c20">L</font> <a href="vectsp_6.html#R1" title="VECTSP_6:pred.1">=</a> <font color="Maroon" title="c21">K1</font> <a href="vectsp_6.html#K5" title="VECTSP_6:func.5">+</a> <font color="Maroon" title="c16">K2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E20:28"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E34:28"><span class="lab"><font color="Green" title="E41">A26</font></span></a>, <a class="txt" href="#E9:28"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E15:28"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E24:28"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E30:28"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="matrlin.html#T6" title="MATRLIN:th.6">MATRLIN:6</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">k</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="c22">k</font> &amp; <font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:28_2"/><span class="lab"><font color="Green" title="E51">A36</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">k</font> &amp; <font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font></span><br/><a NAME="E2:28_2"/><span class="lab"><font color="Green" title="E52">A37</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:28"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="txt" href="#E28:28"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E26:28"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E39:28"><span class="lab"><font color="Green" title="E46">A31</font></span></a>, <a class="txt" href="#E40:28"><span class="lab"><font color="Green" title="E47">A32</font></span></a>, <a class="txt" href="#E38:28"><span class="lab"><font color="Green" title="E45">A30</font></span></a>, <a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E1:28_2"><span class="lab"><font color="Green" title="E51">A36</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_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="E1:28_2_1_1"/>
( <font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> or  ex <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span> &amp; <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_2"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_2_1_1_1"/><span class="lab"><font color="Green" title="E53">A38</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font></span><br/><div class="add"><a NAME="E2:28_2_1_1_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">k</font> &amp; <font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A39</font></span>: <font color="Maroon" title="c19">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:28"><span class="lab"><font color="Green" title="E31">A16</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_2_1_1_1"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_2_1_1_1"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">b1k</font> = <font color="Maroon" title="c5">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:28"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="ref" href="vectsp_4.html#T10" title="VECTSP_4:th.10">VECTSP_4:10</a></span>;<br/><a NAME="E5:28_2_1_1_1"/><span class="lab"><font color="Green" title="E55">A40</font></span>: 
<font color="Maroon" title="c21">K1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:28"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/><a NAME="E6:28_2_1_1_1"/>
 not <font color="Maroon" title="c5">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">K2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:28_2_1_1_1_2"/><span class="lab"><font color="Green" title="E56">A41</font></span>: 
<font color="Maroon" title="c5">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</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/>
<span class="kw">assume </span><a NAME="E2:28_2_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A42</font></span>: 
<font color="Maroon" title="c5">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c16">K2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:28_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E25">A10</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="E4:28_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_2_1_1_1_2"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="ref" href="vectsp_5.html#T3" title="VECTSP_5:th.3">VECTSP_5:3</a></span>;<br/>
<a NAME="E5:28_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E16">A1</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="E6:28_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c5">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T11" title="VECTSP_4:th.11">VECTSP_4:11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:28_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="txt" href="#E30:28"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="txt" href="#E2:28_2_1_1_1_2"><span class="lab"><font color="Green" title="E57">A42</font></span></a>, <a class="ref" href="vectsp_7.html#T2" title="VECTSP_7:th.2">VECTSP_7:2</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><span class="kw">end;</span></div><a NAME="E7:28_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">K2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">b1k</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">K</font>
 ;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A43</font></span>: <font color="Maroon" title="c20">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">b1k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">K1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">b1k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:28"><span class="lab"><font color="Green" title="E50">A35</font></span></a>, <a class="ref" href="vectsp_6.html#T22" title="VECTSP_6:th.22">VECTSP_6:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_2_1_1_1"><span class="lab"><font color="Green" title="E54">A39</font></span></a>, <a class="txt" href="#E5:28_2_1_1_1"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a></span>
;<br/><font color="Maroon" title="c23">b1k</font> = 
<font color="Maroon" title="c5">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:28"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="txt" href="#E1:28_2_1_1_1"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E37:28"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="txt" href="#E1:28_2_1_1_1"><span class="lab"><font color="Green" title="E53">A38</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:28"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="txt" href="#E2:28_2"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E1:28_2"><span class="lab"><font color="Green" title="E51">A36</font></span></a>, <a class="txt" href="#E8:28_2_1_1_1"><span class="lab"><font color="Green" title="E56">A43</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:28"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="txt" href="#E27:28"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E2:28_2"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_2_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span> &amp; <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:28_2_1_1_2"/><span class="lab"><font color="Green" title="E53">A44</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:28_2_1_1_2"/><span class="lab"><font color="Green" title="E54">A45</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c23">n</font>
 ;<br/><a NAME="E5:28_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_2_1_1_2"><span class="lab"><font color="Green" title="E53">A44</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A46</font></span>: <font color="Maroon" title="c15">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:28"><span class="lab"><font color="Green" title="E23">A8</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_2_1_1_2"><span class="lab"><font color="Green" title="E53">A44</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_2_1_1_2"><span class="lab"><font color="Green" title="E53">A44</font></span></a>, <a class="txt" href="#E4:28_2_1_1_2"><span class="lab"><font color="Green" title="E54">A45</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">b2n</font> = <font color="Maroon" title="c6">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="vectsp_4.html#T10" title="VECTSP_4:th.10">VECTSP_4:10</a></span>;<br/><a NAME="E8:28_2_1_1_2"/><span class="lab"><font color="Green" title="E56">A47</font></span>: 
<font color="Maroon" title="c16">K2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:28"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/><a NAME="E9:28_2_1_1_2"/>
 not <font color="Maroon" title="c6">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c21">K1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:28_2_1_1_2_2"/><span class="lab"><font color="Green" title="E57">A48</font></span>: 
<font color="Maroon" title="c6">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c21">K1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:28_2_1_1_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c6">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font> &amp; <font color="Maroon" title="c6">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:28"><span class="lab"><font color="Green" title="E40">A25</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="E3:28_2_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K2" title="VECTSP_5:func.2">/\</a> <font color="Maroon" title="c4">W2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#T3" title="VECTSP_5:th.3">VECTSP_5:3</a></span>;<br/>
<a NAME="E4:28_2_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E16">A1</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="E5:28_2_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c6">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W1</font> <a href="vectsp_5.html#K1" title="VECTSP_5:func.1">+</a> <font color="Maroon" title="c4">W2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T11" title="VECTSP_4:th.11">VECTSP_4:11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:28_2_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:28"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="txt" href="#E30:28"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="txt" href="#E1:28_2_1_1_2_2"><span class="lab"><font color="Green" title="E57">A48</font></span></a>, <a class="ref" href="vectsp_7.html#T2" title="VECTSP_7:th.2">VECTSP_7:2</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><span class="kw">end;</span></div><a NAME="E10:28_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">K1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">b2n</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">K</font>
 ;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A49</font></span>: <font color="Maroon" title="c20">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">b2n</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">K2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">b2n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:28"><span class="lab"><font color="Green" title="E50">A35</font></span></a>, <a class="ref" href="vectsp_6.html#T22" title="VECTSP_6:th.22">VECTSP_6:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28_2_1_1_2"><span class="lab"><font color="Green" title="E55">A46</font></span></a>, <a class="txt" href="#E8:28_2_1_1_2"><span class="lab"><font color="Green" title="E56">A47</font></span></a>, <a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a></span>
;<br/><font color="Maroon" title="c24">b2n</font> = 
<font color="Maroon" title="c6">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:28"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E3:28_2_1_1_2"><span class="lab"><font color="Green" title="E53">A44</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E36:28"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="txt" href="#E29:28"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E3:28_2_1_1_2"><span class="lab"><font color="Green" title="E53">A44</font></span></a>, <a class="txt" href="#E4:28_2_1_1_2"><span class="lab"><font color="Green" title="E54">A45</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:28"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="txt" href="#E2:28_2"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E1:28_2"><span class="lab"><font color="Green" title="E51">A36</font></span></a>, <a class="txt" href="#E11:28_2_1_1_2"><span class="lab"><font color="Green" title="E57">A49</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:28"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="txt" href="#E27:28"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E2:28_2"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/> <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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:28_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
 ; <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">hence </span><a NAME="E46:28"/>
<font color="Maroon" title="c9">v</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">w1</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">w2</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:28"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="txt" href="#E28:28"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="txt" href="#E26:28"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E39:28"><span class="lab"><font color="Green" title="E46">A31</font></span></a>, <a class="txt" href="#E38:28"><span class="lab"><font color="Green" title="E45">A30</font></span></a>, <a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="matrlin2.html#T21" target="_self" title="MATRLIN2:th.21">Th21</a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</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>
