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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/>  for <font color="Olive" title="b2">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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span><br/>  for <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">q2</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> 2</span>)</span><br/>  for <font color="Olive" title="b5">F</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b6">s</font> where <font color="Olive" title="b6">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b3">p2</font>,<font color="Olive" title="b1">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b4">q2</font>,<font color="Olive" title="b1">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">s</font> ) </span> } </span>  holds <br/>( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b5">F</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>  implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Olive" title="b2">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b3">p2</font>,<font color="Olive" title="b1">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Olive" title="b2">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b4">q2</font>,<font color="Olive" title="b1">K</font>)</span>)</span>) ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b5">F</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 implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Olive" title="b2">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b3">p2</font>,<font color="Olive" title="b1">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Olive" title="b2">X</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b4">q2</font>,<font color="Olive" title="b1">K</font>)</span>)</span>)</span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span><br/>  for <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">q2</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> 2</span>)</span><br/>  for <font color="Olive" title="b4">F</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b5">s</font> where <font color="Olive" title="b5">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b3">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">s</font> ) </span> } </span>  holds <br/>( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b4">F</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>  implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Olive" title="b1">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Olive" title="b1">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b3">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b4">F</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 implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Olive" title="b1">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Olive" title="b1">X</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b3">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be    <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></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">  for <font color="Olive" title="b1">p2</font>, <font color="Olive" title="b2">q2</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> 2</span>)</span><br/>  for <font color="Olive" title="b3">F</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b4">s</font> where <font color="Olive" title="b4">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b1">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b2">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">s</font> ) </span> } </span>  holds <br/>( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b3">F</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>  implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b1">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b2">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b3">F</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 implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b1">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Olive" title="b2">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">p2</font>, <font color="Maroon" title="c5">q2</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> 2</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">F</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">s</font> where <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font> ) </span> } </span>  holds <br/>( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">F</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>  implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">F</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 implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">F</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <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="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> ) </span> } </span>  implies ( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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>  implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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 implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span> ) ) )</span><br/>











<span class="kw">assume </span><a NAME="E1:12"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> : ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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>  implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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 implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">Pq</font> =  <a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">Pp</font> =  <a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c9">2S</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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E2:12"/>
<font color="Maroon" title="c3">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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E3:12"/><span class="kw">then </span>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">F</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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Y</font> = <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E5:12"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font></span><br/>

<a NAME="E2:12_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> st <br/>( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:12_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>;<br/>
<a NAME="E7:12"/>
<font color="Maroon" title="c3">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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E8:12"/><span class="kw">then </span>
<font color="Maroon" title="c6">F</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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">F9</font> = <font color="Maroon" title="c6">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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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/>
<span class="kw">set </span><font color="Maroon" title="c12">KK</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">mm</font> =  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">Gp</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E11:12"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c11">F9</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Gp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">and </span><br/><a NAME="E12:12"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b1">e</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> holds <br/><font color="Maroon" title="c14">Gp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font>
 <span class="kw">and </span><br/><a NAME="E13:12"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> holds  <font color="Maroon" title="c14">Gp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">and </span><br/><a NAME="E14:12"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
 for <font color="Olive" title="b1">B</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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span>  st <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Olive" title="b1">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">F9</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c14">Gp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setwiseo.html#D3" title="SETWISEO:def.3">SETWISEO:def 3</a></span>;<br/>
<a NAME="E15:12"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c10">Y</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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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/>
<span class="kw">consider </span><font color="Maroon" title="c15">Gq</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E17:12"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c11">F9</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">and </span><br/><a NAME="E18:12"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b1">e</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> holds <br/><font color="Maroon" title="c15">Gq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font>
 <span class="kw">and </span><br/><a NAME="E19:12"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> holds  <font color="Maroon" title="c15">Gq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">and </span><br/><a NAME="E20:12"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
 for <font color="Olive" title="b1">B</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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span>  st <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Olive" title="b1">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c15">Gq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</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="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setwiseo.html#D3" title="SETWISEO:def.3">SETWISEO:def 3</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">B</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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</font> holds <br/>( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B</font></span>)</span> ) );<br/>
<a NAME="E21:12"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
 for <font color="Olive" title="b1">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> holds <br/><span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</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="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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"> ( <font color="Maroon" title="c16">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> implies <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s</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="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_2"/>
<font color="Maroon" title="c16">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</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"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s</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="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s</font></span>)</span></span><br/>

<a NAME="E2:12_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A14</font></span>: 
 ex <font color="Olive" title="b1">s9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">s9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s</font> &amp; <font color="Olive" title="b1">s9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>;<br/>
<a NAME="E3:12_2"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s</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="c2">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="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s</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="c2">K</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T5" target="_self" title="MATRIX11:th.5">Th5</a></span>;<br/>
<a NAME="E4:12_2"/>
( <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s</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="c2">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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s</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="c2">K</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T5" target="_self" title="MATRIX11:th.5">Th5</a></span>;<br/>
<a NAME="E5:12_2"/><span class="kw">then </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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_2"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E3:12_2"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:12_2"/>
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s</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="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T16" title="VECTSP_1:th.16">VECTSP_1:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E22:12"/><span class="lab"><font color="Green" title="E22">A16</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">k</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_3"/><span class="lab"><font color="Green" title="E23">A17</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c17">k1</font> = <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<span class="kw">let </span><font color="Maroon" title="c18">B</font> be    <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></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"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</font> implies ( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font></span>)</span> ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:12_3"/><span class="lab"><font color="Green" title="E24">A18</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E3:12_3"/><span class="lab"><font color="Green" title="E25">A19</font></span>: 
<font color="Maroon" title="c18">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</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"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font></span>)</span> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_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="E1:12_3_1_1"/>
( <font color="Maroon" title="c16">k</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="c16">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:12_3_1_1_1"/><span class="lab"><font color="Green" title="E26">A20</font></span>: 
<font color="Maroon" title="c16">k</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"> ( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font></span>)</span> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:12_3_1_1_1"/><span class="lab"><font color="Green" title="E27">A21</font></span>: 
<font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c19">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_3"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="ref" href="card_2.html#T42" title="CARD_2:th.42">CARD_2:42</a></span>;<br/><a NAME="E4:12_3_1_1_1"/><span class="lab"><font color="Green" title="E28">A22</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_3_1_1_1"><span class="lab"><font color="Green" title="E27">A21</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E5:12_3_1_1_1"/>
<font color="Maroon" title="c18">B</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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c19">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_3_1_1_1"><span class="lab"><font color="Green" title="E28">A22</font></span></a></span>;<br/><a NAME="E7:12_3_1_1_1"/><span class="lab"><font color="Green" title="E29">A23</font></span>: 
<font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</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="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:12"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E3:12_3_1_1_1"><span class="lab"><font color="Green" title="E27">A21</font></span></a></span>;<br/><a NAME="E8:12_3_1_1_1"/>
<font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:12"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E3:12_3_1_1_1"><span class="lab"><font color="Green" title="E27">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:12_3_1_1_1"/>
( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:12"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E2:12_3"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="txt" href="#E3:12_3"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="txt" href="#E1:12_3_1_1_1"><span class="lab"><font color="Green" title="E26">A20</font></span></a>, <a class="txt" href="#E4:12_3_1_1_1"><span class="lab"><font color="Green" title="E28">A22</font></span></a>, <a class="txt" href="#E7:12_3_1_1_1"><span class="lab"><font color="Green" title="E29">A23</font></span></a>, <a class="ref" href="nat_d.html#T14" title="NAT_D:th.14">NAT_D:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:12_3_1_1_2"/><span class="lab"><font color="Green" title="E26">A24</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font></span>)</span> ) )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c19">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:12_3_1_1_2"/><span class="lab"><font color="Green" title="E27">A25</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_3"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E4:12_3_1_1_2"/>
<font color="Maroon" title="c18">B</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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c19">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_3_1_1_2"><span class="lab"><font color="Green" title="E27">A25</font></span></a></span>;<br/><a NAME="E6:12_3_1_1_2"/>
<font color="Maroon" title="c18">B</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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E7:12_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">B</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">B9</font> = <font color="Maroon" title="c18">B</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> 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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E9:12_3_1_1_2"/><span class="lab"><font color="Green" title="E28">A26</font></span>: 
 not <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">B9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></span>;<br/><a NAME="E10:12_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A27</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c21">B9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_3"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="txt" href="#E3:12_3_1_1_2"><span class="lab"><font color="Green" title="E27">A25</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E11:12_3_1_1_2"/><span class="lab"><font color="Green" title="E30">A28</font></span>: 
<font color="Maroon" title="c21">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_3"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E12:12_3_1_1_2"/><span class="lab"><font color="Green" title="E31">A29</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c21">B9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_3_1_1_2"><span class="lab"><font color="Green" title="E27">A25</font></span></a>, <a class="ref" href="zfmisc_1.html#T116" title="ZFMISC_1:th.116">ZFMISC_1:116</a></span>;<br/><a NAME="E13:12_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A30</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c21">B9</font></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="#E2:12_3"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="txt" href="#E9:12_3_1_1_2"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="ref" href="card_2.html#T41" title="CARD_2:th.41">CARD_2:41</a></span>;<br/><a NAME="E14:12_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A31</font></span>: 
<font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">B9</font></span>)</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="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:12"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E3:12_3"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="txt" href="#E1:12_3_1_1_2"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="txt" href="#E12:12_3_1_1_2"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="txt" href="#E10:12_3_1_1_2"><span class="lab"><font color="Green" title="E29">A27</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E15:12_3_1_1_2"/><span class="lab"><font color="Green" title="E34">A32</font></span>: 
<font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">B9</font></span>)</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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:12"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E3:12_3"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="txt" href="#E1:12_3_1_1_2"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="txt" href="#E12:12_3_1_1_2"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="txt" href="#E13:12_3_1_1_2"><span class="lab"><font color="Green" title="E32">A30</font></span></a>, <a class="txt" href="#E10:12_3_1_1_2"><span class="lab"><font color="Green" title="E29">A27</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_1_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="E1:12_3_1_1_2_1_1"/>
( <font color="Maroon" title="c16">k</font> <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 <font color="Maroon" title="c16">k</font> <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="12_3_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:12_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E35">A33</font></span>: 
<font color="Maroon" title="c16">k</font> <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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font></span>)</span> ) )</span><br/><div class="add"><a NAME="E2:12_3_1_1_2_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 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E3:12_3_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A34</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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 href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E35">A33</font></span></a>, <a class="ref" href="radix_1.html#T2" title="RADIX_1:th.2">RADIX_1:2</a></span>;<br/><a NAME="E4:12_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E37">A35</font></span>: 
<font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">B9</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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"><span class="p3">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:12"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E3:12_3"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="txt" href="#E3:12_3_1_1_2"><span class="lab"><font color="Green" title="E27">A25</font></span></a>, <a class="txt" href="#E15:12_3_1_1_2"><span class="lab"><font color="Green" title="E34">A32</font></span></a></span>;<br/><a NAME="E5:12_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">B9</font></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#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_3"><span class="lab"><font color="Green" title="E23">A17</font></span></a>, <a class="txt" href="#E13:12_3_1_1_2"><span class="lab"><font color="Green" title="E32">A30</font></span></a>, <a class="txt" href="#E11:12_3_1_1_2"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="txt" href="#E14:12_3_1_1_2"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="txt" href="#E1:12_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E35">A33</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:12_3_1_1_2_1_1_1"/>
( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_3"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="txt" href="#E4:12_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E37">A35</font></span></a>, <a class="txt" href="#E3:12_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E36">A34</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_3_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:12_3_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E35">A36</font></span>: 
<font color="Maroon" title="c16">k</font> <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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font></span>)</span> ) )</span><br/><div class="add"><a NAME="E2:12_3_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E36">A37</font></span>: 
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">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"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:12"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E3:12_3"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="txt" href="#E3:12_3_1_1_2"><span class="lab"><font color="Green" title="E27">A25</font></span></a></span>;<br/><a NAME="E3:12_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">B9</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">B9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_3"><span class="lab"><font color="Green" title="E23">A17</font></span></a>, <a class="txt" href="#E13:12_3_1_1_2"><span class="lab"><font color="Green" title="E32">A30</font></span></a>, <a class="txt" href="#E11:12_3_1_1_2"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="txt" href="#E1:12_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E35">A36</font></span></a></span>;<br/><a NAME="E4:12_3_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A38</font></span>: 
<font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.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"><font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">B9</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:12"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E3:12_3"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="txt" href="#E1:12_3_1_1_2"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="txt" href="#E12:12_3_1_1_2"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="txt" href="#E13:12_3_1_1_2"><span class="lab"><font color="Green" title="E32">A30</font></span></a>, <a class="txt" href="#E10:12_3_1_1_2"><span class="lab"><font color="Green" title="E29">A27</font></span></a>, <a class="txt" href="#E2:12_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E5:12_3_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E38">A39</font></span>: 
2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E6:12_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">B9</font></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#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:12"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E3:12_3"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="txt" href="#E1:12_3_1_1_2"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="txt" href="#E12:12_3_1_1_2"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="txt" href="#E13:12_3_1_1_2"><span class="lab"><font color="Green" title="E32">A30</font></span></a>, <a class="txt" href="#E10:12_3_1_1_2"><span class="lab"><font color="Green" title="E29">A27</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:12_3_1_1_2_1_1_2"/>
( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_3"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="txt" href="#E1:12_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="txt" href="#E4:12_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E37">A38</font></span></a>, <a class="txt" href="#E5:12_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="ref" href="radix_1.html#T1" title="RADIX_1:th.1">RADIX_1:1</a>, <a class="ref" href="vectsp_1.html#T10" title="VECTSP_1:th.10">VECTSP_1:10</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><span class="kw">hence </span><a NAME="E17:12_3_1_1_2"/>
( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:12_3"/>
( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">B</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:12"/><span class="lab"><font color="Green" title="E23">A40</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">B</font> be    <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></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"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c16">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</font> implies ( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">B</font></span>)</span> ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:12_4"/><span class="lab"><font color="Green" title="E24">A41</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:12_4"/>
<font color="Maroon" title="c16">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</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"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">B</font></span>)</span> ) )</span><br/>

<a NAME="E3:12_4"/><span class="lab"><font color="Green" title="E25">A42</font></span>: 
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T26" title="NAT_D:th.26">NAT_D:26</a></span>;<br/>
<a NAME="E4:12_4"/><span class="lab"><font color="Green" title="E26">A43</font></span>: 
<font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_4"><span class="lab"><font color="Green" title="E24">A41</font></span></a></span>;<br/>
<a NAME="E5:12_4"/><span class="kw">then </span>
<font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">B</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="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:12"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="fvsum_1.html#T4" title="FVSUM_1:th.4">FVSUM_1:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:12_4"/>
( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">B</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>  implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">B</font> ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">B</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 implies <font color="Maroon" title="c14">Gp</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">B</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="c15">Gq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">B</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:12"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E4:12_4"><span class="lab"><font color="Green" title="E26">A43</font></span></a>, <a class="txt" href="#E3:12_4"><span class="lab"><font color="Green" title="E25">A42</font></span></a>, <a class="ref" href="fvsum_1.html#T4" title="FVSUM_1:th.4">FVSUM_1:4</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>
<a NAME="E24:12"/><span class="lab"><font color="Green" title="E24">A44</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E23:12"><span class="lab"><font color="Green" title="E23">A40</font></span></a>, <a class="txt" href="#E22:12"><span class="lab"><font color="Green" title="E22">A16</font></span></a>);<br/></span>
<a NAME="E25:12"/><span class="lab"><font color="Green" title="E25">A45</font></span>: 
<font color="Maroon" title="c10">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">F</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="E26:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A46</font></span>: 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c10">Y</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c11">F9</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="setwop_2.html#T4" title="SETWOP_2:th.4">SETWOP_2:4</a></span>;<br/>
<a NAME="E27:12"/><span class="lab"><font color="Green" title="E27">A47</font></span>: 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c10">Y</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c11">F9</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:12"><span class="lab"><font color="Green" title="E25">A45</font></span></a>, <a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="setwop_2.html#T4" title="SETWOP_2:th.4">SETWOP_2:4</a></span>;<br/>
<a NAME="E28:12"/><span class="lab"><font color="Green" title="E28">A48</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E29:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A49</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E30:12"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E31:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A50</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E32:12"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></span>)</span> holds <br/><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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</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="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</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="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_5"/><span class="lab"><font color="Green" title="E31">A51</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</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="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font></span><br/>

<a NAME="E2:12_5"/>
<font color="Maroon" title="c10">Y</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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">x9</font> = <font color="Maroon" title="c16">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:12"><span class="lab"><font color="Green" title="E29">A49</font></span></a>, <a class="txt" href="#E1:12_5"><span class="lab"><font color="Green" title="E31">A51</font></span></a></span>;<br/>
<a NAME="E4:12_5"/><span class="lab"><font color="Green" title="E32">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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x9</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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_5"><span class="lab"><font color="Green" title="E31">A51</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E5:12_5"/><span class="lab"><font color="Green" title="E33">A53</font></span>: 
 not <font color="Maroon" title="c17">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:12"><span class="lab"><font color="Green" title="E29">A49</font></span></a>, <a class="txt" href="#E1:12_5"><span class="lab"><font color="Green" title="E31">A51</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:12_5"/><span class="lab"><font color="Green" title="E34">A54</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="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:12_5"/>
<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="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x9</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="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:12"><span class="lab"><font color="Green" title="E29">A49</font></span></a>, <a class="txt" href="#E31:12"><span class="lab"><font color="Green" title="E30">A50</font></span></a>, <a class="txt" href="#E1:12_5"><span class="lab"><font color="Green" title="E31">A51</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:12_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E29:12"><span class="lab"><font color="Green" title="E29">A49</font></span></a>, <a class="txt" href="#E1:12_5"><span class="lab"><font color="Green" title="E31">A51</font></span></a>, <a class="txt" href="#E6:12_5"><span class="lab"><font color="Green" title="E34">A54</font></span></a>, <a class="txt" href="#E4:12_5"><span class="lab"><font color="Green" title="E32">A52</font></span></a>, <a class="txt" href="#E5:12_5"><span class="lab"><font color="Green" title="E33">A53</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E33:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A55</font></span>: 
<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</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="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:12"><span class="lab"><font color="Green" title="E28">A48</font></span></a>, <a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E31:12"><span class="lab"><font color="Green" title="E30">A50</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E34:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A56</font></span>: 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c10">Y</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c10">Y</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setwop_2.html#T7" title="SETWOP_2:th.7">SETWOP_2:7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_6"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_6_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="E1:12_6_1"/>
( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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="12_6_1_1"><span class="kw">case </span></a><a NAME="E1:12_6_1_1"/>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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>  implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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 implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span> ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:12_6_1_1"/>
( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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>  implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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 implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E17:12"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E24:12"><span class="lab"><font color="Green" title="E24">A44</font></span></a>, <a class="txt" href="#E34:12"><span class="lab"><font color="Green" title="E32">A56</font></span></a>, <a class="txt" href="#E26:12"><span class="lab"><font color="Green" title="E26">A46</font></span></a>, <a class="txt" href="#E27:12"><span class="lab"><font color="Green" title="E27">A47</font></span></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="12_6_1_2"><span class="kw">case </span></a><a NAME="E1:12_6_1_2"/><span class="lab"><font color="Green" title="E33">A57</font></span>: 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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>  implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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 implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span> ) )</span><br/><div class="add"><a NAME="E2:12_6_1_2"/><span class="lab"><font color="Green" title="E34">A58</font></span>: 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c10">Y</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c11">F9</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:12"><span class="lab"><font color="Green" title="E31">A55</font></span></a>, <a class="txt" href="#E27:12"><span class="lab"><font color="Green" title="E27">A47</font></span></a>, <a class="ref" href="setwop_2.html#T7" title="SETWOP_2:th.7">SETWOP_2:7</a></span>;<br/><a NAME="E3:12_6_1_2"/>
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c10">Y</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c11">F9</font>,<span class="p3">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E17:12"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E24:12"><span class="lab"><font color="Green" title="E24">A44</font></span></a>, <a class="txt" href="#E26:12"><span class="lab"><font color="Green" title="E26">A46</font></span></a>, <a class="txt" href="#E1:12_6_1_2"><span class="lab"><font color="Green" title="E33">A57</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:12_6_1_2"/>
( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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>  implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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 implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>)</span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_6_1_2"><span class="lab"><font color="Green" title="E33">A57</font></span></a>, <a class="txt" href="#E2:12_6_1_2"><span class="lab"><font color="Green" title="E34">A58</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>
<span class="kw">hence </span><a NAME="E36:12"/>
( ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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>  implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) ) &amp; ( <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">F</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 implies  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c2">K</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><a href="matrix11.html#K1" title="MATRIX11:func.1">Part_sgn</a> (<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c2">K</font>)</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"> verum</span><br/>


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