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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p1</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span><br/>  for <font color="Olive" title="b2">K</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/>  for <font color="Olive" title="b3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">K</font><br/>  for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j</font> holds <br/> <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="Olive" title="b2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b2">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b5">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Olive" title="b3">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b4">i</font>)</span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font> be    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/>  for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">K</font><br/>  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j</font> holds <br/> <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Olive" title="b2">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="Olive" title="b1">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b1">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b4">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Olive" title="b2">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Olive" title="b3">i</font>)</span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font><br/>  for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">j</font> holds <br/> <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Olive" title="b1">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Olive" title="b2">i</font>)</span>)</span>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> holds <br/> <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Olive" title="b1">i</font>)</span>)</span>)</span>)</span></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">n1</font> = <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> implies  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>)</span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:26"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>)</span>)</span></span><br/>

<a NAME="E3:26"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c2">p1</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D9" title="MATRIX_2:def.9">MATRIX_2:def 9</a></span>;<br/>
<a NAME="E4:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E5:26"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/>
<a NAME="E6:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">R</font> =  <a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>);<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T23" title="NAT_1:th.23">NAT_1:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>)</span>)</span></span><br/><div class="add"><a NAME="E2:26_1_1"/><span class="kw">then </span>
 <a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>) is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="laplace.html#T11" target="_self" title="LAPLACE:th.11">Th11</a></span>;<br/><a NAME="E3:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A7</font></span>: 
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></span>;<br/><a NAME="E4:26_1_1"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1
 ;<br/><a NAME="E5:26_1_1"/>
<font color="Maroon" title="c2">p1</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="laplace.html#T11" target="_self" title="LAPLACE:th.11">Th11</a></span>;<br/><a NAME="E6:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A9</font></span>: 
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></span>;<br/><a NAME="E7:26_1_1"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E8:26_1_1"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E9:26_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_1"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="hurwitz.html#T4" title="HURWITZ:th.4">HURWITZ:4</a></span>;<br/><span class="kw">hence </span><a NAME="E10:26_1_1"/>
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_1_1"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E3:26_1_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="vectsp_1.html#D4" title="VECTSP_1:def.4">VECTSP_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_2"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>)</span>)</span></span><br/><div class="add"><a NAME="E2:26_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A12</font></span>: 
<font color="Maroon" title="c2">p1</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D9" title="MATRIX_2:def.9">MATRIX_2:def 9</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_2_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">1,2</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> or <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">2,1</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_2"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="matrix_7.html#T1" title="MATRIX_7:th.1">MATRIX_7:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_2_1_1"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">1,2</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>)</span>)</span></span><br/><div class="add"><a NAME="E2:26_1_2_1_1"/>
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:26_1_2"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E3:26_1_2_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_2_1_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E4:26_1_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 or <font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a></span>;<br/><a NAME="E5:26_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A14</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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T4" title="HURWITZ:th.4">HURWITZ:4</a></span>;<br/><a NAME="E6:26_1_2_1_1"/><span class="lab"><font color="Green" title="E31">A15</font></span>: 
 <a href="matrix_2.html#K11" title="MATRIX_2:func.11">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>;<br/><a NAME="E7:26_1_2_1_1"/>
 <a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>) is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_2"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="laplace.html#T11" target="_self" title="LAPLACE:th.11">Th11</a></span>;<br/><a NAME="E8:26_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A16</font></span>: 
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></span>;<br/><a NAME="E9:26_1_2_1_1"/>
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2</span>)</span> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T25" title="MATRIX_2:th.25">MATRIX_2:25</a></span>;<br/><a NAME="E10:26_1_2_1_1"/><span class="kw">then </span>
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_2"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E1:26_1_2_1_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E6:26_1_2_1_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="finseq_2.html#T52" title="FINSEQ_2:th.52">FINSEQ_2:52</a>, <a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E11:26_1_2_1_1"/>
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_2_1_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E8:26_1_2_1_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="vectsp_1.html#D4" title="VECTSP_1:def.4">VECTSP_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_2_1_2"/><span class="lab"><font color="Green" title="E29">A17</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">2,1</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>)</span>)</span></span><br/><div class="add"><a NAME="E2:26_1_2_1_2"/>
 <a href="matrix_2.html#K11" title="MATRIX_2:func.11">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>;<br/><a NAME="E3:26_1_2_1_2"/><span class="kw">then </span>
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_2"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E1:26_1_2_1_2"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a>, <a class="ref" href="matrix_9.html#T12" title="MATRIX_9:th.12">MATRIX_9:12</a></span>;<br/><a NAME="E4:26_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A18</font></span>: 
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D4" title="VECTSP_1:def.4">VECTSP_1:def 4</a></span>;<br/><a NAME="E5:26_1_2_1_2"/>
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:26_1_2"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E6:26_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E1:26_1_2_1_2"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E7:26_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A19</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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T4" title="HURWITZ:th.4">HURWITZ:4</a></span>;<br/><a NAME="E8:26_1_2_1_2"/>
 <a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>) is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_2"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="laplace.html#T11" target="_self" title="LAPLACE:th.11">Th11</a></span>;<br/><a NAME="E9:26_1_2_1_2"/><span class="kw">then </span>
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E10:26_1_2_1_2"/>
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_2_1_2"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E4:26_1_2_1_2"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="ref" href="vectsp_1.html#T8" title="VECTSP_1:th.8">VECTSP_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3"><span class="kw">suppose </span></a><a NAME="E1:26_1_3"/><span class="lab"><font color="Green" title="E27">A20</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>)</span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">n2</font> = <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_3_1"/>
(  not <font color="Maroon" title="c3">K</font> is  <a href="vectsp_1.html#V12" title="VECTSP_1:attr.12">Fanoian</a>  or <font color="Maroon" title="c3">K</font> is  <a href="vectsp_1.html#V12" title="VECTSP_1:attr.12">Fanoian</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_1"/><span class="lab"><font color="Green" title="E28">A21</font></span>: 
 not <font color="Maroon" title="c3">K</font> is  <a href="vectsp_1.html#V12" title="VECTSP_1:attr.12">Fanoian</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>)</span>)</span></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E29">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_3_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_3_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T12" title="NAT_D:th.12">NAT_D:12</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:26_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E30">A23</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">t</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>
 <span class="kw">and </span><br/><a NAME="E4:26_1_3_1_1_1_1_1"/>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D2" title="NAT_D:def.2">NAT_D:def 2</a></span>;<br/><a NAME="E5:26_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c10">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:26_1_3_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="ref" href="hurwitz.html#T4" title="HURWITZ:th.4">HURWITZ:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:26_1_3_1_1_1_1_2"/><span class="lab"><font color="Green" title="E30">A24</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">t</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E4:26_1_3_1_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D2" title="NAT_D:def.2">NAT_D:def 2</a></span>;<br/><a NAME="E5:26_1_3_1_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A25</font></span>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_1"><span class="lab"><font color="Green" title="E28">A21</font></span></a>, <a class="ref" href="matrix11.html#T22" title="MATRIX11:th.22">MATRIX11:22</a></span>;<br/><a NAME="E6:26_1_3_1_1_1_1_2"/>
<font color="Maroon" title="c10">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E7:26_1_3_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E30">A24</font></span></a>, <a class="txt" href="#E5:26_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A25</font></span></a>, <a class="ref" href="hurwitz.html#T4" title="HURWITZ:th.4">HURWITZ:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E3:26_1_3_1_1"/><span class="lab"><font color="Green" title="E30">A26</font></span>: 
(  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> or  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></span>;<br/><a NAME="E4:26_1_3_1_1"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_1"><span class="lab"><font color="Green" title="E28">A21</font></span></a>, <a class="ref" href="matrix11.html#T22" title="MATRIX11:th.22">MATRIX11:22</a></span>;<br/><a NAME="E5:26_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A27</font></span>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T9" title="VECTSP_1:th.9">VECTSP_1:9</a></span>;<br/><a NAME="E6:26_1_3_1_1"/><span class="lab"><font color="Green" title="E32">A28</font></span>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D4" title="VECTSP_1:def.4">VECTSP_1:def 4</a></span>;<br/><a NAME="E7:26_1_3_1_1"/>
(  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> or  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></span>;<br/><a NAME="E8:26_1_3_1_1"/><span class="kw">then </span>
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p1">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_1_3_1_1"><span class="lab"><font color="Green" title="E32">A28</font></span></a>, <a class="txt" href="#E5:26_1_3_1_1"><span class="lab"><font color="Green" title="E31">A27</font></span></a>, <a class="ref" href="vectsp_1.html#D4" title="VECTSP_1:def.4">VECTSP_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E9:26_1_3_1_1"/>
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_3_1_1"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="txt" href="#E3:26_1_3_1_1"><span class="lab"><font color="Green" title="E30">A26</font></span></a>, <a class="txt" href="#E5:26_1_3_1_1"><span class="lab"><font color="Green" title="E31">A27</font></span></a>, <a class="ref" href="vectsp_1.html#D4" title="VECTSP_1:def.4">VECTSP_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_2"/><span class="lab"><font color="Green" title="E28">A29</font></span>: 
<font color="Maroon" title="c3">K</font> is  <a href="vectsp_1.html#V12" title="VECTSP_1:attr.12">Fanoian</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>)</span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">mm</font> =  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">K</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">n1</font> = <font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">set </span><font color="Maroon" title="c12">P1</font> =  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">Q1</font> = <font color="Maroon" title="c2">p1</font> as    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c14">SS1</font> =  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">X</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:26_1_3_1_2"/><span class="lab"><font color="Green" title="E29">A30</font></span>: 
<font color="Maroon" title="c15">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">N</font>,<font color="Maroon" title="c6">i</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> where <font color="Olive" title="b1">N</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">N</font>,<font color="Maroon" title="c6">i</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E6:26_1_3_1_2"/><span class="lab"><font color="Green" title="E30">A31</font></span>: 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c15">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span>) <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2"><span class="lab"><font color="Green" title="E28">A29</font></span></a>, <a class="ref" href="laplace.html#T9" target="_self" title="LAPLACE:th.9">Th9</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">PQ1</font> =  <a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>);<br/><span class="kw">set </span><font color="Maroon" title="c17">SS2</font> =  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">Q19</font> = <font color="Maroon" title="c13">Q1</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D9" title="MATRIX_2:def.9">MATRIX_2:def 9</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">P2</font> =  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">Q</font> =  <a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>) as    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">Q9</font> = <font color="Maroon" title="c20">Q</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D9" title="MATRIX_2:def.9">MATRIX_2:def 9</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c22">PQ</font> =  <a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>);<br/><a NAME="E10:26_1_3_1_2"/><span class="lab"><font color="Green" title="E31">A32</font></span>: 
 <a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D14" title="MATRIX_2:def.14">MATRIX_2:def 14</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">SSX</font> = <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">X</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/><a NAME="E12:26_1_3_1_2"/><span class="lab"><font color="Green" title="E32">A33</font></span>: 
<font color="Maroon" title="c15">X</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c23">SSX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>;<br/><a NAME="E13:26_1_3_1_2"/>
<font color="Maroon" title="c15">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/><a NAME="E14:26_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A34</font></span>: 
<font color="Maroon" title="c15">X</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c23">SSX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_1_3_1_2"><span class="lab"><font color="Green" title="E32">A33</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E16:26_1_3_1_2"/><span class="lab"><font color="Green" title="E34">A35</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">X</font>
 <span class="kw">and </span><br/><a NAME="E17:26_1_3_1_2"/><span class="lab"><font color="Green" title="E35">A36</font></span>: 
<font color="Maroon" title="c24">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E18:26_1_3_1_2"/><span class="lab"><font color="Green" title="E36">A37</font></span>: 
 for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> holds <br/>( ( <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> implies <font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) &amp; ( <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> implies <font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">k</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) &amp; ( <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> implies <font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E1:26_1_3"><span class="lab"><font color="Green" title="E27">A20</font></span></a>, <a class="txt" href="#E5:26_1_3_1_2"><span class="lab"><font color="Green" title="E29">A30</font></span></a>, <a class="ref" href="laplace.html#T10" target="_self" title="LAPLACE:th.10">Th10</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c25">Pf</font> = <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font>;<br/><a NAME="E19:26_1_3_1_2"/><span class="lab"><font color="Green" title="E37">A38</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E20:26_1_3_1_2"/><span class="lab"><font color="Green" title="E38">A39</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c18">Q19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/><div><span class="lab"><font color="Green" title="E39">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_3_1_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:26_1_3_1_2_1"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E2:26_1_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A41</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E3:26_1_3_1_2_1"/><span class="lab"><font color="Green" title="E41">A42</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><span class="kw">consider </span><font color="Maroon" title="c27">k</font>, <font color="Maroon" title="c28">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:26_1_3_1_2_1"/><span class="lab"><font color="Green" title="E42">A43</font></span>: 
<font color="Maroon" title="c27">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:26_1_3_1_2_1"/><span class="lab"><font color="Green" title="E43">A44</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:26_1_3_1_2_1"/><span class="lab"><font color="Green" title="E44">A45</font></span>: 
<font color="Maroon" title="c27">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">m</font>
 <span class="kw">and </span><br/><a NAME="E8:26_1_3_1_2_1"/><span class="lab"><font color="Green" title="E45">A46</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c27">k</font>,<font color="Maroon" title="c28">m</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="ref" href="matrix11.html#T1" title="MATRIX11:th.1">MATRIX11:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c29">k</font> = <font color="Maroon" title="c27">k</font>, <font color="Maroon" title="c30">m</font> = <font color="Maroon" title="c28">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E10:26_1_3_1_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c21">Q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/><a NAME="E11:26_1_3_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">Q9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><a NAME="E12:26_1_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A47</font></span>: 
( <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> or <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c31">m1</font> = <font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><span class="kw">set </span><font color="Maroon" title="c32">k1</font> = <font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><a NAME="E13:26_1_3_1_2_1"/><span class="lab"><font color="Green" title="E47">A48</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ;<br/><a NAME="E14:26_1_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A49</font></span>: 
<font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></span>;<br/><a NAME="E15:26_1_3_1_2_1"/><span class="lab"><font color="Green" title="E49">A50</font></span>: 
<font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E13:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E47">A48</font></span></a>, <a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_3_1_2_1_1"/>
( ( <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> ) or ( <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> ) or ( <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> ) or ( <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E50">A51</font></span>: 
( <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:26_1_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E51">A52</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:26_1_3_1_2"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E3:26_1_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E52">A53</font></span>: 
<font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26_1_3_1_2"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E50">A51</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_3_1_2_1_1_1_1"/>
( ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> ) or ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> ) or ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> ) or ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_2_1_1_1_1_1"/>
( ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> ) or ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:26_1_3_1_2_1_1_1_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> &amp; <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> ) or ( <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E3:26_1_3_1_2_1_1_1_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> ) or ( <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E46">A47</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E4:26_1_3_1_2_1_1_1_1_1"/><span class="kw">then </span>
( ( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font> ) or ( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:26_1_3_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:26_1_3_1_2"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E52">A53</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A54</font></span>: 
( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:26_1_3_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E3:26_1_3_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A55</font></span>: 
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E4:26_1_3_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c18">Q19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E20:26_1_3_1_2"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><a NAME="E5:26_1_3_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:26_1_3_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A56</font></span>: 
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_3_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E54">A55</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:26_1_3_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:26_1_3_1_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A57</font></span>: 
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>;<br/><a NAME="E9:26_1_3_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E10:26_1_3_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A56</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E11:26_1_3_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E52">A53</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E51">A52</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_2_1_1_1_1_3"/><span class="lab"><font color="Green" title="E53">A58</font></span>: 
( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:26_1_3_1_2_1_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E3:26_1_3_1_2_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A59</font></span>: 
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E4:26_1_3_1_2_1_1_1_1_3"/>
<font color="Maroon" title="c18">Q19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E20:26_1_3_1_2"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><a NAME="E5:26_1_3_1_2_1_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E53">A58</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:26_1_3_1_2_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A60</font></span>: 
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_3_1_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E54">A59</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:26_1_3_1_2_1_1_1_1_3"/>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E53">A58</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:26_1_3_1_2_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A61</font></span>: 
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>;<br/><a NAME="E9:26_1_3_1_2_1_1_1_1_3"/>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E53">A58</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E10:26_1_3_1_2_1_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E53">A58</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E55">A60</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E11:26_1_3_1_2_1_1_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E52">A53</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E51">A52</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E56">A61</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_2_1_1_2"/>
( <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:26_1_3_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="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="26_1_3_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_2_1_1_3"/><span class="lab"><font color="Green" title="E50">A62</font></span>: 
( <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:26_1_3_1_2_1_1_3"/><span class="lab"><font color="Green" title="E51">A63</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c29">k</font>,<font color="Maroon" title="c30">m</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:26_1_3_1_2"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E3:26_1_3_1_2_1_1_3"/><span class="lab"><font color="Green" title="E52">A64</font></span>: 
<font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c29">k</font>,<font color="Maroon" title="c30">m</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c29">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26_1_3_1_2"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_3"><span class="lab"><font color="Green" title="E50">A62</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_3_1_2_1_1_3_1"/>
( ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> ) or ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> ) or ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> ) or ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_2_1_1_3_1_1"/>
( ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> ) or ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:26_1_3_1_2_1_1_3_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> &amp; <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_3"><span class="lab"><font color="Green" title="E50">A62</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E3:26_1_3_1_2_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A65</font></span>: 
( ( <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E46">A47</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E4:26_1_3_1_2_1_1_3_1_1"/>
<font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:26_1_3_1_2_1_1_3_1_1"/><span class="kw">then </span>
( ( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c29">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font> ) or ( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c29">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="txt" href="#E15:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E53">A65</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:26_1_3_1_2_1_1_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:26_1_3_1_2"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1_1_3"><span class="lab"><font color="Green" title="E52">A64</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_2_1_1_3_1_2"/><span class="lab"><font color="Green" title="E53">A66</font></span>: 
( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:26_1_3_1_2_1_1_3_1_2"/>
<font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_3"><span class="lab"><font color="Green" title="E50">A62</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:26_1_3_1_2_1_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">Q19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E20:26_1_3_1_2"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="txt" href="#E15:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><a NAME="E4:26_1_3_1_2_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A67</font></span>: 
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E53">A66</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:26_1_3_1_2_1_1_3_1_2"/>
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_3"><span class="lab"><font color="Green" title="E50">A62</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E53">A66</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E6:26_1_3_1_2_1_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E7:26_1_3_1_2_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A68</font></span>: 
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_3_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E54">A67</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:26_1_3_1_2_1_1_3_1_2"/>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_3"><span class="lab"><font color="Green" title="E50">A62</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E53">A66</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E9:26_1_3_1_2_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A69</font></span>: 
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E53">A66</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E55">A68</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:26_1_3_1_2_1_1_3_1_2"/><span class="lab"><font color="Green" title="E57">A70</font></span>: 
<font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E11:26_1_3_1_2_1_1_3_1_2"/>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E53">A66</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:26_1_3_1_2_1_1_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c29">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="txt" href="#E15:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="txt" href="#E10:26_1_3_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E57">A70</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E13:26_1_3_1_2_1_1_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1_1_3"><span class="lab"><font color="Green" title="E52">A64</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1_1_3"><span class="lab"><font color="Green" title="E51">A63</font></span></a>, <a class="txt" href="#E9:26_1_3_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E56">A69</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_2_1_1_3_1_3"/><span class="lab"><font color="Green" title="E53">A71</font></span>: 
( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:26_1_3_1_2_1_1_3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_3"><span class="lab"><font color="Green" title="E50">A62</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E3:26_1_3_1_2_1_1_3_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A72</font></span>: 
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E4:26_1_3_1_2_1_1_3_1_3"/>
<font color="Maroon" title="c18">Q19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E20:26_1_3_1_2"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_3"><span class="lab"><font color="Green" title="E50">A62</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><a NAME="E5:26_1_3_1_2_1_1_3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_3_1_3"><span class="lab"><font color="Green" title="E53">A71</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:26_1_3_1_2_1_1_3_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A73</font></span>: 
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_3_1_2_1_1_3_1_3"><span class="lab"><font color="Green" title="E54">A72</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:26_1_3_1_2_1_1_3_1_3"/>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_3"><span class="lab"><font color="Green" title="E50">A62</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_3_1_3"><span class="lab"><font color="Green" title="E53">A71</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E8:26_1_3_1_2_1_1_3_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A74</font></span>: 
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_3_1_3"><span class="lab"><font color="Green" title="E53">A71</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1_1_3_1_3"><span class="lab"><font color="Green" title="E55">A73</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:26_1_3_1_2_1_1_3_1_3"/><span class="lab"><font color="Green" title="E57">A75</font></span>: 
<font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E10:26_1_3_1_2_1_1_3_1_3"/>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_3_1_3"><span class="lab"><font color="Green" title="E53">A71</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:26_1_3_1_2_1_1_3_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c29">k</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="txt" href="#E15:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="txt" href="#E9:26_1_3_1_2_1_1_3_1_3"><span class="lab"><font color="Green" title="E57">A75</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:26_1_3_1_2_1_1_3_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1_1_3"><span class="lab"><font color="Green" title="E52">A64</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1_1_3"><span class="lab"><font color="Green" title="E51">A63</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1_1_3_1_3"><span class="lab"><font color="Green" title="E56">A74</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_2_1_1_4"/><span class="lab"><font color="Green" title="E50">A76</font></span>: 
( <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:26_1_3_1_2_1_1_4"/><span class="lab"><font color="Green" title="E51">A77</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c29">k</font>,<font color="Maroon" title="c30">m</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:26_1_3_1_2"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E3:26_1_3_1_2_1_1_4"/><span class="lab"><font color="Green" title="E52">A78</font></span>: 
<font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E4:26_1_3_1_2_1_1_4"/><span class="lab"><font color="Green" title="E53">A79</font></span>: 
<font color="Maroon" title="c24">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c29">k</font>,<font color="Maroon" title="c30">m</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26_1_3_1_2"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E50">A76</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_3_1_2_1_1_4_1"/>
( ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> ) or ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> ) or ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> ) or ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_2_1_1_4_1_1"/>
( ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> ) or ( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:26_1_3_1_2_1_1_4_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E50">A76</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E3:26_1_3_1_2_1_1_4_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E46">A47</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E4:26_1_3_1_2_1_1_4_1_1"/><span class="kw">then </span>
( ( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font> ) or ( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E14:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E48">A49</font></span></a>, <a class="txt" href="#E15:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E52">A78</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:26_1_3_1_2_1_1_4_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:26_1_3_1_2"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E4:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E53">A79</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_2_1_1_4_1_2"/><span class="lab"><font color="Green" title="E54">A80</font></span>: 
( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:26_1_3_1_2_1_1_4_1_2"/>
<font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E50">A76</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:26_1_3_1_2_1_1_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">Q19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E20:26_1_3_1_2"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="txt" href="#E15:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><a NAME="E4:26_1_3_1_2_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A81</font></span>: 
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_4_1_2"><span class="lab"><font color="Green" title="E54">A80</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:26_1_3_1_2_1_1_4_1_2"/>
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E50">A76</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_4_1_2"><span class="lab"><font color="Green" title="E54">A80</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E6:26_1_3_1_2_1_1_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E7:26_1_3_1_2_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A82</font></span>: 
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_3_1_2_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A81</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:26_1_3_1_2_1_1_4_1_2"/>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_4_1_2"><span class="lab"><font color="Green" title="E54">A80</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:26_1_3_1_2_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A83</font></span>: 
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E48">A49</font></span></a>, <a class="txt" href="#E15:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E52">A78</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>;<br/><a NAME="E10:26_1_3_1_2_1_1_4_1_2"/>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E50">A76</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_4_1_2"><span class="lab"><font color="Green" title="E54">A80</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E11:26_1_3_1_2_1_1_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_4_1_2"><span class="lab"><font color="Green" title="E54">A80</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1_1_4_1_2"><span class="lab"><font color="Green" title="E56">A82</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E12:26_1_3_1_2_1_1_4_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E4:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E53">A79</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E51">A77</font></span></a>, <a class="txt" href="#E9:26_1_3_1_2_1_1_4_1_2"><span class="lab"><font color="Green" title="E57">A83</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3_1_2_1_1_4_1_3"><span class="kw">suppose </span></a><a NAME="E1:26_1_3_1_2_1_1_4_1_3"/><span class="lab"><font color="Green" title="E54">A84</font></span>: 
( <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:26_1_3_1_2_1_1_4_1_3"/>
<font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E50">A76</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:26_1_3_1_2_1_1_4_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c18">Q19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E20:26_1_3_1_2"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="txt" href="#E14:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E48">A49</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/><a NAME="E4:26_1_3_1_2_1_1_4_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A85</font></span>: 
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_4_1_3"><span class="lab"><font color="Green" title="E54">A84</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:26_1_3_1_2_1_1_4_1_3"/>
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E50">A76</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_4_1_3"><span class="lab"><font color="Green" title="E54">A84</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E6:26_1_3_1_2_1_1_4_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E7:26_1_3_1_2_1_1_4_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A86</font></span>: 
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_3_1_2_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A85</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:26_1_3_1_2_1_1_4_1_3"/>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_4_1_3"><span class="lab"><font color="Green" title="E54">A84</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:26_1_3_1_2_1_1_4_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A87</font></span>: 
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E48">A49</font></span></a>, <a class="txt" href="#E15:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="txt" href="#E3:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E52">A78</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>;<br/><a NAME="E10:26_1_3_1_2_1_1_4_1_3"/>
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E50">A76</font></span></a>, <a class="txt" href="#E1:26_1_3_1_2_1_1_4_1_3"><span class="lab"><font color="Green" title="E54">A84</font></span></a>, <a class="ref" href="laplace.html#D3" target="_self" title="LAPLACE:def.3">Def3</a></span>;<br/><a NAME="E11:26_1_3_1_2_1_1_4_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_3_1_2_1_1_4_1_3"><span class="lab"><font color="Green" title="E54">A84</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1_1_4_1_3"><span class="lab"><font color="Green" title="E56">A86</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E12:26_1_3_1_2_1_1_4_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E6:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E7:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="txt" href="#E8:26_1_3_1_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="txt" href="#E4:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E53">A79</font></span></a>, <a class="txt" href="#E2:26_1_3_1_2_1_1_4"><span class="lab"><font color="Green" title="E51">A77</font></span></a>, <a class="txt" href="#E9:26_1_3_1_2_1_1_4_1_3"><span class="lab"><font color="Green" title="E57">A87</font></span></a>, <a class="ref" href="matrix11.html#D1" title="MATRIX11:def.1">MATRIX11:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">reconsider </span><font color="Maroon" title="c26">domf</font> =  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c24">f</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/><a NAME="E23:26_1_3_1_2"/><span class="lab"><font color="Green" title="E40">A88</font></span>: 
<font color="Maroon" title="c24">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c26">domf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T113" title="RELAT_1:th.113">RELAT_1:113</a></span>;<br/><a NAME="E24:26_1_3_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c24">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E25:26_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A89</font></span>: 
<font color="Maroon" title="c26">domf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D14" title="MATRIX_2:def.14">MATRIX_2:def 14</a></span>;<br/><a NAME="E26:26_1_3_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E27:26_1_3_1_2"/><span class="kw">then </span>
 <a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c24">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:26_1_3_1_2"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="txt" href="#E21:26_1_3_1_2"><span class="lab"><font color="Green" title="E39">A40</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>;<br/><a NAME="E28:26_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A90</font></span>: 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c23">SSX</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:26_1_3_1_2"><span class="lab"><font color="Green" title="E34">A35</font></span></a>, <a class="txt" href="#E17:26_1_3_1_2"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="txt" href="#E25:26_1_3_1_2"><span class="lab"><font color="Green" title="E41">A89</font></span></a>, <a class="txt" href="#E23:26_1_3_1_2"><span class="lab"><font color="Green" title="E40">A88</font></span></a>, <a class="ref" href="setwop_2.html#T6" title="SETWOP_2:th.6">SETWOP_2:6</a></span>;<br/><a NAME="E29:26_1_3_1_2"/>
<font color="Maroon" title="c15">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">SSX</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/><a NAME="E30:26_1_3_1_2"/><span class="kw">then </span>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> (<font color="Maroon" title="c13">Q1</font>,<font color="Maroon" title="c3">K</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_1_3_1_2"><span class="lab"><font color="Green" title="E30">A31</font></span></a>, <a class="txt" href="#E28:26_1_3_1_2"><span class="lab"><font color="Green" title="E42">A90</font></span></a>, <a class="txt" href="#E14:26_1_3_1_2"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="txt" href="#E10:26_1_3_1_2"><span class="lab"><font color="Green" title="E31">A32</font></span></a>, <a class="ref" href="setwop_2.html#T4" title="SETWOP_2:th.4">SETWOP_2:4</a></span>;<br/><span class="kw">hence </span> <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p1</font>) = 
<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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T26" title="MATRIX11:th.26">MATRIX11:26</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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> (<font color="Maroon" title="c20">Q</font>,<font color="Maroon" title="c3">K</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1: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="c3">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> (<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c6">i</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T26" title="MATRIX11:th.26">MATRIX11:26</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
