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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 holds <br/> <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">i</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 holds <br/> <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">i</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">i</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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 implies  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:4"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_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="E1:4_1"/>
( <font color="Maroon" title="c3">i</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> <font color="Maroon" title="c2">A</font> or  not <font color="Maroon" title="c3">i</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> <font color="Maroon" title="c2">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><span class="kw">suppose </span></a><a NAME="E1:4_1_1"/>
<font color="Maroon" title="c3">i</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> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:4_1_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E4:4_1_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a></span>;<br/><a NAME="E5:4_1_1"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E3:4_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c4">m</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"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:4_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E8:4_1_1"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font></span>)</span> <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="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T115" title="FINSEQ_3:th.115">FINSEQ_3:115</a></span>;<br/><a NAME="E9:4_1_1"/><span class="lab"><font color="Green" title="E12">A8</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font></span>)</span>,<font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4_1_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>;<br/><a NAME="E10:4_1_1"/>
<font color="Maroon" title="c2">A</font> is   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">A</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></span>;<br/><a NAME="E11:4_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font></span>)</span>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:4_1_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E9:4_1_1"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="txt" href="#E8:4_1_1"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E12:4_1_1"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:4_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E5:4_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E7:4_1_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E9:4_1_1"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2"><span class="kw">suppose </span></a><a NAME="E1:4_1_2"/>
not <font color="Maroon" title="c3">i</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> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:4_1_2"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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