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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b1">V1</font><br/>  for <font color="Olive" title="b3">L1</font>, <font color="Olive" title="b4">L2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Scalar</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b2">f</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>  &amp; <font color="Olive" title="b3">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">L2</font> &amp; <font color="Olive" title="b4">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b2">f</font> holds <br/> for <font color="Olive" title="b5">I</font> being    <a href="vectsp_5.html#M1" title="VECTSP_5:mode.1">Linear_Compl</a> of  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b3">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Olive" title="b1">V1</font></span>)</span></span>)</span></span>)</span><br/>  for <font color="Olive" title="b6">fI</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b5">I</font>,<font color="Olive" title="b5">I</font>  st <font color="Olive" title="b6">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <font color="Olive" title="b5">I</font> holds <br/>( <font color="Olive" title="b6">fI</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>  &amp; <font color="Olive" title="b3">L1</font> is  not    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b6">fI</font> &amp; <font color="Olive" title="b4">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b6">fI</font> &amp;  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b4">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Olive" title="b1">V1</font></span>)</span></span>)</span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b5">I</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V1</font> be   <a href="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c2">V1</font><br/>  for <font color="Olive" title="b2">L1</font>, <font color="Olive" title="b3">L2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Scalar</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">f</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>  &amp; <font color="Olive" title="b2">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">L2</font> &amp; <font color="Olive" title="b3">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">f</font> holds <br/> for <font color="Olive" title="b4">I</font> being    <a href="vectsp_5.html#M1" title="VECTSP_5:mode.1">Linear_Compl</a> of  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b2">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span><br/>  for <font color="Olive" title="b5">fI</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b4">I</font>,<font color="Olive" title="b4">I</font>  st <font color="Olive" title="b5">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <font color="Olive" title="b4">I</font> holds <br/>( <font color="Olive" title="b5">fI</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>  &amp; <font color="Olive" title="b2">L1</font> is  not    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b5">fI</font> &amp; <font color="Olive" title="b3">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b5">fI</font> &amp;  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b3">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b4">I</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c2">V1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L1</font>, <font color="Olive" title="b2">L2</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Scalar</a> of <font color="Maroon" title="c1">K</font>  st <font color="Maroon" title="c3">f</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>  &amp; <font color="Olive" title="b1">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">L2</font> &amp; <font color="Olive" title="b2">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c3">f</font> holds <br/> for <font color="Olive" title="b3">I</font> being    <a href="vectsp_5.html#M1" title="VECTSP_5:mode.1">Linear_Compl</a> of  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span><br/>  for <font color="Olive" title="b4">fI</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b3">I</font>,<font color="Olive" title="b3">I</font>  st <font color="Olive" title="b4">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <font color="Olive" title="b3">I</font> holds <br/>( <font color="Olive" title="b4">fI</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>  &amp; <font color="Olive" title="b1">L1</font> is  not    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b4">fI</font> &amp; <font color="Olive" title="b2">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b4">fI</font> &amp;  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b2">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b3">I</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">L1</font>, <font color="Maroon" title="c5">L2</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Scalar</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>  &amp; <font color="Maroon" title="c4">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">L2</font> &amp; <font color="Maroon" title="c5">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c3">f</font> implies  for <font color="Olive" title="b1">I</font> being    <a href="vectsp_5.html#M1" title="VECTSP_5:mode.1">Linear_Compl</a> of  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span><br/>  for <font color="Olive" title="b2">fI</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font>  st <font color="Olive" title="b2">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <font color="Olive" title="b1">I</font> holds <br/>( <font color="Olive" title="b2">fI</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>  &amp; <font color="Maroon" title="c4">L1</font> is  not    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b2">fI</font> &amp; <font color="Maroon" title="c5">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b2">fI</font> &amp;  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">I</font> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> 
 <span class="kw">and </span><br/><a NAME="E2:57"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
<font color="Maroon" title="c4">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">L2</font>
 <span class="kw">and </span><br/><a NAME="E3:57"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
<font color="Maroon" title="c5">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">I</font> being    <a href="vectsp_5.html#M1" title="VECTSP_5:mode.1">Linear_Compl</a> of  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span><br/>  for <font color="Olive" title="b2">fI</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b1">I</font>,<font color="Olive" title="b1">I</font>  st <font color="Olive" title="b2">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <font color="Olive" title="b1">I</font> holds <br/>( <font color="Olive" title="b2">fI</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>  &amp; <font color="Maroon" title="c4">L1</font> is  not    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b2">fI</font> &amp; <font color="Maroon" title="c5">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b2">fI</font> &amp;  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">I</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">V</font> = <font color="Maroon" title="c2">V1</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V1</font><span class="kw"> such that </span><br/><a NAME="E5:57"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font>
 <span class="kw">and </span><br/><a NAME="E6:57"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L2</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c7">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="vectsp11.html#D2" target="_self" onmouseover="rs('vectsp11/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">U</font> =  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">fU</font> = <font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span> as   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T31" target="_self" onmouseover="rs('vectsp11/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">f2</font> = <font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">I</font> be    <a href="vectsp_5.html#M1" title="VECTSP_5:mode.1">Linear_Compl</a> of  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">fI</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c12">I</font>,<font color="Maroon" title="c12">I</font>  st <font color="Olive" title="b1">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <font color="Maroon" title="c12">I</font> holds <br/>( <font color="Olive" title="b1">fI</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>  &amp; <font color="Maroon" title="c4">L1</font> is  not    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">fI</font> &amp; <font color="Maroon" title="c5">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Olive" title="b1">fI</font> &amp;  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c12">I</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c13">fI</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c12">I</font>,<font color="Maroon" title="c12">I</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <font color="Maroon" title="c12">I</font> implies ( <font color="Maroon" title="c13">fI</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>  &amp; <font color="Maroon" title="c4">L1</font> is  not    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c13">fI</font> &amp; <font color="Maroon" title="c5">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c13">fI</font> &amp;  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c12">I</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E8:57"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
<font color="Maroon" title="c13">fI</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="matrlin2.html#K8" title="MATRLIN2:func.8">|</a> <font color="Maroon" title="c12">I</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">fI</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a>  &amp; <font color="Maroon" title="c4">L1</font> is  not    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c13">fI</font> &amp; <font color="Maroon" title="c5">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c13">fI</font> &amp;  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c12">I</font> )</span><br/>

<div><span class="lab"><font color="Green" title="E48">A7</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V1</font>  st <font color="Olive" title="b1">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">v</font> is   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c12">I</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">v</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> implies <font color="Maroon" title="c14">v</font> is   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c12">I</font> )</span><br/><span class="kw">assume </span><a NAME="E1:57_1"/>
<font color="Maroon" title="c14">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">v</font> is   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c12">I</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:57_1"/><span class="lab"><font color="Green" title="E49">A8</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">v</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">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T24" target="_self" onmouseover="rs('vectsp11/T24')" onmouseout="rh()">Th24</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">v1</font> = <font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>);<br/><span class="kw">set </span><font color="Maroon" title="c17">i1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p2">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c18">u1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p2">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> ;<br/><a NAME="E4:57_1"/><span class="lab"><font color="Green" title="E50">A9</font></span>: 
<font color="Maroon" title="c2">V1</font> <a href="vectsp_5.html#R1" title="VECTSP_5:pred.1">is_the_direct_sum_of</a> <font color="Maroon" title="c12">I</font>, <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D5" onmouseover="rs('vectsp_5/D5')" onmouseout="rh()">VECTSP_5:def 5</a></span>;<br/><a NAME="E5:57_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A10</font></span>: 
<font color="Maroon" title="c14">v</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="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p3">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p3">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D6" onmouseover="rs('vectsp_5/D6')" onmouseout="rh()">VECTSP_5:def 6</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> $1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p3">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <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">V1</font>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">W</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">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I</font> or <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> ) holds <br/> for <font color="Olive" title="b2">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V1</font>  st <font color="Olive" title="b2">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">W</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> $1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">W</font>;<br/><span class="kw">set </span><font color="Maroon" title="c19">L21</font> = <font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c20">f21</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span>;<br/><a NAME="E6:57_1"/><span class="lab"><font color="Green" title="E52">A11</font></span>: 
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">I</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T17" onmouseover="rs('vectsp_4/T17')" onmouseout="rh()">VECTSP_4:17</a></span>;<br/><a NAME="E7:57_1"/><span class="lab"><font color="Green" title="E53">A12</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:57_1_1"/><span class="lab"><font color="Green" title="E54">A13</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c22">W</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c22">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I</font> or <font color="Maroon" title="c22">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> ) implies  for <font color="Olive" title="b1">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V1</font>  st <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:57_1_1"/><span class="lab"><font color="Green" title="E55">A14</font></span>: 
( <font color="Maroon" title="c22">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I</font> or <font color="Maroon" title="c22">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V1</font>  st <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c23">w</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:57_1_1"/><span class="lab"><font color="Green" title="E56">A15</font></span>: 
<font color="Maroon" title="c23">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c24">fnw</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font>;<br/>
<a NAME="E4:57_1_1"/><span class="lab"><font color="Green" title="E57">A16</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_1"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="txt" href="#E2:57_1_1"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="txt" href="#E3:57_1_1"><span class="lab"><font color="Green" title="E56">A15</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E58">A17</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:57_1_1_1_1"/>
( <font color="Maroon" title="c22">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I</font> or <font color="Maroon" title="c22">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_1_1"><span class="lab"><font color="Green" title="E55">A14</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:57_1_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A18</font></span>: 
<font color="Maroon" title="c22">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">F</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font> as   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c12">I</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1"><span class="lab"><font color="Green" title="E57">A16</font></span></a></span>;<br/><a NAME="E3:57_1_1_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">fI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/><span class="kw">hence </span><a NAME="E4:57_1_1_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A18</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="57_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:57_1_1_1_1_2"/><span class="lab"><font color="Green" title="E59">A19</font></span>: 
<font color="Maroon" title="c22">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">F</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font> as   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1"><span class="lab"><font color="Green" title="E57">A16</font></span></a></span>;<br/><a NAME="E3:57_1_1_1_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">fU</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/><span class="kw">hence </span><a NAME="E4:57_1_1_1_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_1_1_1_1_2"><span class="lab"><font color="Green" title="E59">A19</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D4" onmouseover="rs('matrlin/D4')" onmouseout="rh()">MATRLIN:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span>

;<br/>
<a NAME="E7:57_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A20</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1_1"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="ref" href="vectsp_4.html#T21" onmouseover="rs('vectsp_4/T21')" onmouseout="rh()">VECTSP_4:21</a></span>;<br/>
<a NAME="E8:57_1_1"/><span class="lab"><font color="Green" title="E60">A21</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> 1</span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T20" target="_self" onmouseover="rs('vectsp11/T20')" onmouseout="rh()">Th20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1_1"><span class="lab"><font color="Green" title="E60">A21</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T19" target="_self" onmouseover="rs('vectsp11/T19')" onmouseout="rh()">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D3" onmouseover="rs('matrlin/D3')" onmouseout="rh()">MATRLIN:def 3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:57_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57_1_1"><span class="lab"><font color="Green" title="E58">A17</font></span></a>, <a class="txt" href="#E7:57_1_1"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="ref" href="vectsp_4.html#T20" onmouseover="rs('vectsp_4/T20')" onmouseout="rh()">VECTSP_4:20</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="lab"><font color="Green" title="E54">A22</font></span>: <span class="p1">(<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> <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="c2">V1</font></span>)</span> = 
 <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="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p4">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p4">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57_1"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="txt" href="#E3:57_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="vectsp_1.html#D20" onmouseover="rs('vectsp_1/D20')" onmouseout="rh()">VECTSP_1:def 20</a></span>
;<br/><a NAME="E9:57_1"/><span class="lab"><font color="Green" title="E55">A23</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p2">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="ref" href="vectsp_5.html#D6" onmouseover="rs('vectsp_5/D6')" onmouseout="rh()">VECTSP_5:def 6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E11:57_1"/><span class="lab"><font color="Green" title="E56">A24</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p3">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <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">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T24" target="_self" onmouseover="rs('vectsp11/T24')" onmouseout="rh()">Th24</a></span>;<br/><a NAME="E12:57_1"/><span class="lab"><font color="Green" title="E57">A25</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">W</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c22">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I</font> or <font color="Maroon" title="c22">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> ) implies  for <font color="Olive" title="b1">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V1</font>  st <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:57_1_3"/>
( <font color="Maroon" title="c22">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I</font> or <font color="Maroon" title="c22">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V1</font>  st <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c23">w</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:57_1_3"/><span class="lab"><font color="Green" title="E58">A26</font></span>: 
<font color="Maroon" title="c23">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font></span><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T18" target="_self" onmouseover="rs('vectsp11/T18')" onmouseout="rh()">Th18</a></span>
<br/>.= 
<font color="Maroon" title="c23">w</font>

;<br/>
<span class="kw">hence </span><a NAME="E4:57_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c22">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_1_3"><span class="lab"><font color="Green" title="E58">A26</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E13:57_1"/><span class="lab"><font color="Green" title="E58">A27</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E12:57_1"><span class="lab"><font color="Green" title="E57">A25</font></span></a>, <a class="txt" href="#E7:57_1"><span class="lab"><font color="Green" title="E53">A12</font></span></a>);<br/></span><a NAME="E14:57_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A28</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p3">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_1"><span class="lab"><font color="Green" title="E55">A23</font></span></a></span>;<br/><a NAME="E15:57_1"/><span class="lab"><font color="Green" title="E60">A29</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p2">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="ref" href="vectsp_5.html#D6" onmouseover="rs('vectsp_5/D6')" onmouseout="rh()">VECTSP_5:def 6</a></span>;<br/><a NAME="E16:57_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p3">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57_1"><span class="lab"><font color="Green" title="E58">A27</font></span></a></span>;<br/><a NAME="E17:57_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p3">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <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">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_1"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E14:57_1"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E6:57_1"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="txt" href="#E8:57_1"><span class="lab"><font color="Green" title="E54">A22</font></span></a>, <a class="ref" href="vectsp_5.html#T48" onmouseover="rs('vectsp_5/T48')" onmouseout="rh()">VECTSP_5:48</a></span>;<br/><a NAME="E18:57_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A30</font></span>: 
 ex <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c22">MIN</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E20:57_1"/><span class="lab"><font color="Green" title="E62">A31</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c22">MIN</font>]
 <span class="kw">and </span><br/><a NAME="E21:57_1"/><span class="lab"><font color="Green" title="E63">A32</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c22">MIN</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" onmouseover="rs('nat_1/S5')" onmouseout="rh()">NAT_1:sch 5</a>(<a class="txt" href="#E18:57_1"><span class="lab"><font color="Green" title="E61">A30</font></span></a>);<br/></span><span class="kw">assume </span><a NAME="E22:57_1"/><span class="lab"><font color="Green" title="E64">A33</font></span>: 
<font color="Maroon" title="c14">v</font> is  not   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c12">I</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E23:57_1"/><span class="lab"><font color="Green" title="E65">A34</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p2">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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="txt" href="#E5:57_1"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a>, <a class="txt" href="#E15:57_1"><span class="lab"><font color="Green" title="E60">A29</font></span></a>, <a class="txt" href="#E22:57_1"><span class="lab"><font color="Green" title="E64">A33</font></span></a></span>;<br/><a NAME="E24:57_1"/>
<font color="Maroon" title="c21">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:57_1_4"/>
<font color="Maroon" title="c21">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p3">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57_1"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="ref" href="vectsp11.html#T18" target="_self" onmouseover="rs('vectsp11/T18')" onmouseout="rh()">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p2">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 

;<br/>
<span class="kw">hence </span><a NAME="E3:57_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:57_1"><span class="lab"><font color="Green" title="E65">A34</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">h</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c2">V1</font><span class="kw"> such that </span><br/><a NAME="E26:57_1"/><span class="lab"><font color="Green" title="E66">A35</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <font color="Maroon" title="c23">h</font></span>)</span> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E27:57_1"/><span class="lab"><font color="Green" title="E67">A36</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <font color="Maroon" title="c23">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c23">h</font> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T38" target="_self" onmouseover="rs('vectsp11/T38')" onmouseout="rh()">Th38</a>, <a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><a NAME="E28:57_1"/><span class="lab"><font color="Green" title="E68">A37</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E29:57_1"/>
<font color="Maroon" title="c22">MIN</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:57_1_5"/>
<font color="Maroon" title="c22">MIN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p3">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:57_1"><span class="lab"><font color="Green" title="E62">A31</font></span></a>, <a class="ref" href="vectsp11.html#T18" target="_self" onmouseover="rs('vectsp11/T18')" onmouseout="rh()">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p2">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 

;<br/>
<span class="kw">hence </span><a NAME="E3:57_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:57_1"><span class="lab"><font color="Green" title="E65">A34</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">M1</font> = <font color="Maroon" title="c22">MIN</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" onmouseover="rs('nat_1/T20')" onmouseout="rh()">NAT_1:20</a></span>;<br/><span class="lab"><font color="Green" title="E69">A38</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <font color="Maroon" title="c23">h</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <font color="Maroon" title="c23">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> 1</span>)</span></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <font color="Maroon" title="c23">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T19" target="_self" onmouseover="rs('vectsp11/T19')" onmouseout="rh()">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">M1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <font color="Maroon" title="c23">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T20" target="_self" onmouseover="rs('vectsp11/T20')" onmouseout="rh()">Th20</a></span>
<br/>.= 
<font color="Maroon" title="c23">h</font> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c22">MIN</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:57_1"><span class="lab"><font color="Green" title="E67">A36</font></span></a></span>
;<br/><a NAME="E32:57_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A39</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p3">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p4">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span>,<font color="Maroon" title="c21">n</font>)</span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p4">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E51"><span class="lab"><font color="Green" title="E39">Lm9</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span>,<font color="Maroon" title="c21">n</font>)</span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p5">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D4" onmouseover="rs('matrlin/D4')" onmouseout="rh()">MATRLIN:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span>,<font color="Maroon" title="c21">n</font>)</span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p4">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span>,<font color="Maroon" title="c21">n</font>)</span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p4">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_2.html#D2" onmouseover="rs('mod_2/D2')" onmouseout="rh()">MOD_2:def 2</a></span>
;<br/><a NAME="E34:57_1"/><span class="lab"><font color="Green" title="E71">A40</font></span>: 
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span>,<font color="Maroon" title="c21">n</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:57_1_8"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span>,<font color="Maroon" title="c21">n</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:57_1_8"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#T5" onmouseover="rs('matrixj1/T5')" onmouseout="rh()">MATRIXJ1:5</a></span>;<br/>
<a NAME="E3:57_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A41</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI: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="c21">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T82" onmouseover="rs('fvsum_1/T82')" onmouseout="rh()">FVSUM_1:82</a></span>;<br/>
<a NAME="E4:57_1_8"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c21">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T124" onmouseover="rs('finseq_2/T124')" onmouseout="rh()">FINSEQ_2:124</a></span>;<br/>
<a NAME="E5:57_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_1_8"><span class="lab"><font color="Green" title="E72">A41</font></span></a>, <a class="ref" href="finseq_2.html#T57" onmouseover="rs('finseq_2/T57')" onmouseout="rh()">FINSEQ_2:57</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:57_1_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="vectsp_1.html#T19" onmouseover="rs('vectsp_1/T19')" onmouseout="rh()">VECTSP_1:19</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E35:57_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c22">MIN</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A42</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c23">h</font> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c22">MIN</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p3">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> = 
<font color="Maroon" title="c23">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c22">MIN</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p4">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
 <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="txt" href="#E20:57_1"><span class="lab"><font color="Green" title="E62">A31</font></span></a>, <a class="ref" href="ranknull.html#T9" onmouseover="rs('ranknull/T9')" onmouseout="rh()">RANKNULL:9</a></span>
;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E50"><span class="lab"><font color="Green" title="E38">Lm8</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="vectsp11.html#E52"><span class="lab"><font color="Green" title="E40">Lm10</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T19" onmouseover="rs('vectsp_1/T19')" onmouseout="rh()">VECTSP_1:19</a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a></span>
;<br/><span class="kw">then </span> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p4">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:57_1"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="ref" href="ranknull.html#T9" onmouseover="rs('ranknull/T9')" onmouseout="rh()">RANKNULL:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <font color="Maroon" title="c23">h</font></span>)</span> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p3">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:57_1"><span class="lab"><font color="Green" title="E66">A35</font></span></a>, <a class="txt" href="#E28:57_1"><span class="lab"><font color="Green" title="E68">A37</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h</font> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c22">MIN</font></span>)</span></span>)</span> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="matrlin2.html#K6" title="MATRLIN2:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c21">n</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p3">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:57_1"><span class="lab"><font color="Green" title="E69">A38</font></span></a>, <a class="txt" href="vectsp11.html#E49"><span class="lab"><font color="Green" title="E37">Lm7</font></span></a></span>
<br/>.= 
<span class="p1">(<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> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span>,<font color="Maroon" title="c21">n</font>)</span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p5">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:57_1"><span class="lab"><font color="Green" title="E72">A42</font></span></a>, <a class="txt" href="#E33:57_1"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="ref" href="matrlin.html#D3" onmouseover="rs('matrlin/D3')" onmouseout="rh()">MATRLIN:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">L1</font></span>)</span>,<font color="Maroon" title="c21">n</font>)</span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p5">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p4">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a></span>
;<br/><a NAME="E39:57_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c24">M1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">v</font> <a href="vectsp_5.html#K4" title="VECTSP_5:func.4">|--</a> (<font color="Maroon" title="c12">I</font>,<span class="p3">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">L1</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p0">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>)</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <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">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:57_1"><span class="lab"><font color="Green" title="E71">A40</font></span></a>, <a class="ref" href="vectsp_1.html#T15" onmouseover="rs('vectsp_1/T15')" onmouseout="rh()">VECTSP_1:15</a></span>;<br/><a NAME="E40:57_1"/><span class="kw">then </span>
<font color="Maroon" title="c24">M1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">M1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:57_1"><span class="lab"><font color="Green" title="E63">A32</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E41:57_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E10:57"/>
<font color="Maroon" title="c7">v</font> is    <a href="vectsp11.html#M2" title="VECTSP11:mode.2">eigenvector</a> of <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">L2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="vectsp11.html#D3" target="_self" onmouseover="rs('vectsp11/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E11:57"/><span class="kw">then </span>
<font color="Maroon" title="c7">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="vectsp11.html#T17" target="_self" onmouseover="rs('vectsp11/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<span class="kw">then </span> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T10" onmouseover="rs('ranknull/T10')" onmouseout="rh()">RANKNULL:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T19" target="_self" onmouseover="rs('vectsp11/T19')" onmouseout="rh()">Th19</a></span>
;<br/>
<a NAME="E13:57"/><span class="kw">then </span>
<font color="Maroon" title="c7">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T24" target="_self" onmouseover="rs('vectsp11/T24')" onmouseout="rh()">Th24</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">vI</font> = <font color="Maroon" title="c7">v</font> as   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c12">I</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57"><span class="lab"><font color="Green" title="E48">A7</font></span></a></span>;<br/>
<a NAME="E15:57"/><span class="lab"><font color="Green" title="E49">A43</font></span>: 
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</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="c12">I</font> &amp; <font color="Maroon" title="c5">L2</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L2</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c14">vI</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T11" onmouseover="rs('vectsp_4/T11')" onmouseout="rh()">VECTSP_4:11</a>, <a class="ref" href="vectsp_4.html#T14" onmouseover="rs('vectsp_4/T14')" onmouseout="rh()">VECTSP_4:14</a></span>;<br/>
<a NAME="E16:57"/><span class="lab"><font color="Green" title="E50">A44</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">fI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">vI</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:57"/><span class="lab"><font color="Green" title="E51">A45</font></span>: 
<font color="Maroon" title="c13">fI</font> is  <a href="vectsp11.html#V1" title="VECTSP11:attr.1">with_eigenvalues</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E15:57"><span class="lab"><font color="Green" title="E49">A43</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">L1</font> is  not    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c13">fI</font> &amp; <font color="Maroon" title="c5">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c13">fI</font> &amp;  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c12">I</font> )</span><br/>

<a NAME="E18:57"/>
<font color="Maroon" title="c4">L1</font> is  not    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c13">fI</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:57_3"/>
<font color="Maroon" title="c4">L1</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c13">fI</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c12">I</font><span class="kw"> such that </span><br/><a NAME="E3:57_3"/><span class="lab"><font color="Green" title="E52">A46</font></span>: 
<font color="Maroon" title="c15">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c12">I</font>
 <span class="kw">and </span><br/><a NAME="E4:57_3"/><span class="lab"><font color="Green" title="E53">A47</font></span>: 
<font color="Maroon" title="c13">fI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L1</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c15">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:57"><span class="lab"><font color="Green" title="E51">A45</font></span></a>, <a class="ref" href="vectsp11.html#D2" target="_self" onmouseover="rs('vectsp11/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E5:57_3"/>
<font color="Maroon" title="c15">w</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">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="txt" href="#E4:57_3"><span class="lab"><font color="Green" title="E53">A47</font></span></a>, <a class="ref" href="vectsp11.html#T37" target="_self" onmouseover="rs('vectsp11/T37')" onmouseout="rh()">Th37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:57_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_3"><span class="lab"><font color="Green" title="E52">A46</font></span></a>, <a class="ref" href="vectsp_4.html#T11" onmouseover="rs('vectsp_4/T11')" onmouseout="rh()">VECTSP_4:11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E19:57"/>
( <font color="Maroon" title="c4">L1</font> is  not    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c13">fI</font> &amp; <font color="Maroon" title="c5">L2</font> is    <a href="vectsp11.html#M1" title="VECTSP11:mode.1">eigenvalue</a> of <font color="Maroon" title="c13">fI</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E15:57"><span class="lab"><font color="Green" title="E49">A43</font></span></a>, <a class="txt" href="#E16:57"><span class="lab"><font color="Green" title="E50">A44</font></span></a>, <a class="txt" href="#E17:57"><span class="lab"><font color="Green" title="E51">A45</font></span></a>, <a class="ref" href="vectsp11.html#D2" target="_self" onmouseover="rs('vectsp11/D2')" onmouseout="rh()">Def2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c12">I</font></span><br/>

<a NAME="E20:57"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">I</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">I</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:57_4"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">I</font></span><br/>

<a NAME="E2:57_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A48</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:57_4"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T9" onmouseover="rs('vectsp_4/T9')" onmouseout="rh()">VECTSP_4:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">v</font> = <font color="Maroon" title="c15">x</font> as   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V1</font> ;<br/>
<a NAME="E5:57_4"/>
<font color="Maroon" title="c16">v</font> is   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c12">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="txt" href="#E2:57_4"><span class="lab"><font color="Green" title="E52">A48</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:57_4"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">I</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E21:57"/>
 <a href="vectsp11.html#K2" title="VECTSP11:func.2">UnionKers</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="matrlin.html#K3" title="MATRLIN:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">L2</font></span>)</span> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c12">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T27" onmouseover="rs('vectsp_4/T27')" onmouseout="rh()">VECTSP_4:27</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
