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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/>  for <font color="Olive" title="b2">M</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font> holds <br/> ( <font color="Olive" title="b2">M</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font> iff <font color="Olive" title="b2">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  is   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">M</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> holds <br/> ( <font color="Olive" title="b1">M</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> iff <font color="Olive" title="b1">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  is   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</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">M</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> iff <font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  is   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> )</span><br/>





<span class="kw">thus </span><a NAME="E1:2"/>
( <font color="Maroon" title="c3">M</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> implies <font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  is   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</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">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  is   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> implies <font color="Maroon" title="c3">M</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_1"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c3">M</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</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">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  is   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">j</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"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/><a NAME="E3:2_1_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c4">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><a NAME="E4:2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E2:2_1_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="matrix_2.html#D4" title="MATRIX_2:def.4">MATRIX_2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:2_1"/>
<font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  is   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D3" title="MATRIX_2:def.3">MATRIX_2:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:2"/><span class="lab"><font color="Green" title="E3">A5</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a>  is   <a href="matrix_2.html#NM1" title="MATRIX_2:NM.1">Upper_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</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">M</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_2"/><span class="lab"><font color="Green" title="E4">A6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E2:2_2"/><span class="lab"><font color="Green" title="E5">A7</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</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">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/><a NAME="E3:2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c4">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_2"><span class="lab"><font color="Green" title="E4">A6</font></span></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><a NAME="E4:2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E3">A5</font></span></a>, <a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E5">A7</font></span></a>, <a class="ref" href="matrix_2.html#D3" title="MATRIX_2:def.3">MATRIX_2:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_2"/>
<font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_2"><span class="lab"><font color="Green" title="E4">A6</font></span></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:2"/>
<font color="Maroon" title="c3">M</font> is   <a href="matrix_2.html#NM2" title="MATRIX_2:NM.2">Lower_Triangular_Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D4" title="MATRIX_2:def.4">MATRIX_2:def 4</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>
