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<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V1</font> being <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b1">V1</font><br/> for <font color="Olive" title="b3">J</font> being   <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b4">M</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">b1</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b3">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">b1</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>  holds <br/>for <font color="Olive" title="b5">F</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b1">V1</font>  st <font color="Olive" title="b5">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Olive" title="b4">M</font>,<font color="Olive" title="b2">b1</font>,<font color="Olive" title="b2">b1</font> holds <br/>(  ex <font color="Olive" title="b6">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">J</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b5">F</font> ) &amp; ( for <font color="Olive" title="b6">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">J</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b6">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b5">F</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">V1</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b1</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">V1</font><br/> for <font color="Olive" title="b2">J</font> being   <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b3">M</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">b1</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b2">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">b1</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>  holds <br/>for <font color="Olive" title="b4">F</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c2">V1</font>  st <font color="Olive" title="b4">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Olive" title="b3">M</font>,<font color="Olive" title="b1">b1</font>,<font color="Olive" title="b1">b1</font> holds <br/>(  ex <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">J</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b4">F</font> ) &amp; ( for <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">J</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b4">F</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">b1</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">J</font> being   <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">M</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Olive" title="b1">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</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>  holds <br/>for <font color="Olive" title="b3">F</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c2">V1</font>  st <font color="Olive" title="b3">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Olive" title="b2">M</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c3">b1</font> holds <br/>(  ex <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">J</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b3">F</font> ) &amp; ( for <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">J</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b3">F</font> ) )</span><br/><b>let </b><font color="Maroon" title="c4">J</font> be    <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</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>  holds <br/>for <font color="Olive" title="b2">F</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c2">V1</font>  st <font color="Olive" title="b2">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c3">b1</font> holds <br/>(  ex <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b2">F</font> ) &amp; ( for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b2">F</font> ) )</span><br/><b>let </b><font color="Maroon" title="c5">M</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</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>  implies for <font color="Olive" title="b1">F</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c2">V1</font>  st <font color="Olive" title="b1">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c3">b1</font> holds <br/>(  ex <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b1">F</font> ) &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b1">F</font> ) ) )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green" title="E34">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c2">V1</font>  st <font color="Olive" title="b1">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c3">b1</font> holds <br/>(  ex <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b1">F</font> ) &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Olive" title="b1">F</font> ) )</span><br/>

<b>set </b><font color="Maroon" title="c6">mm</font> =  <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font></span>)</span>;<br/>
<a NAME="E3:48"/><i><font color="Green" title="E35">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:48"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="#E2:48"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E4:48"/><b>then </b><i><font color="Green" title="E36">A4</font></i>: 
(  <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrixj1.html#D1" title="MATRIXJ1:def.1">MATRIXJ1:def 1</a>, <a class="ref" href="matrixj1.html#D3" title="MATRIXJ1:def.3">MATRIXJ1:def 3</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">F</font> be  <a href="matrixj2.html#V2" title="MATRIXJ2:attr.2">nilpotent</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c2">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c3">b1</font> implies (  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font> ) ) )</span><br/>

<b>assume </b><a NAME="E5:48"/><i><font color="Green" title="E37">A5</font></i>: 
<font color="Maroon" title="c7">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c5">M</font>,<font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c3">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font> ) )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:48_1"/><i><font color="Green" title="E38">A6</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span></span><br/><a NAME="E2:48_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="#E4:48"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="#E1:48_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="matrixj1.html#D3" title="MATRIXJ1:def.3">MATRIXJ1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:48_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:48"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="#E1:48_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="polynom3.html#T4" title="POLYNOM3:th.4">POLYNOM3:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E7:48"/><b>then </b><i><font color="Green" title="E38">A7</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c8">MIN</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:48"/><i><font color="Green" title="E39">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">MIN</font>]
 <b>and </b><br/><a NAME="E10:48"/><i><font color="Green" title="E40">A9</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon" title="c8">MIN</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E7:48"><i><font color="Green" title="E38">A7</font></i></a>);<br/></i>
<a NAME="E11:48"/><i><font color="Green" title="E41">A10</font></i>: 
 ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">MIN</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green" title="E42">A11</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">MIN</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:48_2"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <span class="p3">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">MIN</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <span class="p3">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">MIN</font> )
 
<b>by </b><i><a class="txt" href="#E4:48"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="#E9:48"><i><font color="Green" title="E39">A8</font></i></a></i>;<br/>
<a NAME="E3:48_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <span class="p3">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">b1</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">MIN</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">M1</font> = <font color="Maroon" title="c8">MIN</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">M1</font> )</span><br/><b>assume </b><a NAME="E1:48_2_1"/><i><font color="Green" title="E43">A12</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">M1</font></span><br/><a NAME="E2:48_2_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">MIN</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">MIN</font> )
 <b>by </b><i><a class="txt" href="#E9:48"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="#E1:48_2"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="#E1:48_2_1"><i><font color="Green" title="E43">A12</font></i></a></i>;<br/><a NAME="E3:48_2_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">M1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E4:48_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">M1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E6:48_2"/><b>then </b>
<font color="Maroon" title="c9">M1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">M1</font>
 
<b>by </b><i><a class="txt" href="#E10:48"><i><font color="Green" title="E40">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:48_2"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c9">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E13:48"/><i><font color="Green" title="E42">A13</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">MIN</font> )
 ;<br/>
<a NAME="E14:48"/><i><font color="Green" title="E43">A14</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">i</font> )
 
<b>by </b><i><a class="txt" href="#E4:48"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="#E13:48"><i><font color="Green" title="E42">A13</font></i></a>, <a class="ref" href="matrixj1.html#D3" title="MATRIXJ1:def.3">MATRIXJ1:def 3</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">S</font> =  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E15:48"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="txt" href="#E4:48"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="#E13:48"><i><font color="Green" title="E42">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E16:48"/><b>then </b>
(  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font> &amp; <font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a>, <a class="ref" href="polynom3.html#T18" title="POLYNOM3:th.18">POLYNOM3:18</a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>;<br/>
<a NAME="E17:48"/><b>then </b><i><font color="Green" title="E44">A15</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">MIN</font> &amp; $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">MIN</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> $1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> );<br/>
<a NAME="E18:48"/><i><font color="Green" title="E45">A16</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E19:48"/><i><font color="Green" title="E46">A17</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">n</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:48_3"/><i><font color="Green" title="E47">A18</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<b>set </b><font color="Maroon" title="c12">n1</font> = <font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>assume </b><a NAME="E2:48_3"/><i><font color="Green" title="E48">A19</font></i>: 
( <font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">MIN</font> &amp; <font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">MIN</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E3:48_3"/>
<span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">i</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="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">MIN</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="#E4:48"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="#E13:48"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="#E14:48"><i><font color="Green" title="E43">A14</font></i></a>, <a class="txt" href="#E17:48"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="finseq_5.html#T11" title="FINSEQ_5:th.11">FINSEQ_5:11</a></i>;<br/>
<a NAME="E4:48_3"/><b>then </b><i><font color="Green" title="E49">A20</font></i>: 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">MIN</font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>;<br/>
<a NAME="E5:48_3"/>
<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">MIN</font>
 
<b>by </b><i><a class="txt" href="#E2:48_3"><i><font color="Green" title="E48">A19</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:48_3"/><b>then </b><i><font color="Green" title="E50">A21</font></i>: 
( <font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">MIN</font> &amp; <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">MIN</font> )
 
<b>by </b><i><a class="txt" href="#E2:48_3"><i><font color="Green" title="E48">A19</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E7:48_3"/><b>then </b><i><font color="Green" title="E51">A22</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:48_3"><i><font color="Green" title="E49">A20</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:48_3"/>
( <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span> &amp;  <a href="matrixj1.html#K1" title="MATRIXJ1:func.1">min</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font> &amp; <span class="p1">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj2.html#K7" title="MATRIXJ2:func.7">|</a> <font color="Maroon" title="c9">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E4:48"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="#E13:48"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="#E14:48"><i><font color="Green" title="E43">A14</font></i></a>, <a class="txt" href="#E2:48_3"><i><font color="Green" title="E48">A19</font></i></a>, <a class="ref" href="matrixj1.html#T10" title="MATRIXJ1:th.10">MATRIXJ1:10</a>, <a class="ref" href="matrixj1.html#T17" title="MATRIXJ1:th.17">MATRIXJ1:17</a></i>;<br/>
<b>then </b><i><font color="Green" title="E52">A23</font></i>: <font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:48"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="#E3:48"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="#E5:48"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="#E17:48"><i><font color="Green" title="E44">A15</font></i></a>, <a class="txt" href="#E7:48_3"><i><font color="Green" title="E51">A22</font></i></a>, <a class="ref" href="matrixj2.html#T24" target="_self" title="MATRIXJ2:th.24">Th24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T59" title="VECTSP_1:th.59">VECTSP_1:59</a></i>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_3_2_1"/>
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_2_1_1"><b>suppose </b></a><a NAME="E1:48_3_2_1_1"/>
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:48_3_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E9:48_3"><i><font color="Green" title="E52">A23</font></i></a>, <a class="ref" href="vectsp11.html#T19" title="VECTSP11:th.19">VECTSP11:19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_2_1_2"><b>suppose </b></a><a NAME="E1:48_3_2_1_2"/><i><font color="Green" title="E53">A24</font></i>: 
<font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:48_3_2_1_2"/><i><font color="Green" title="E54">A25</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> 1</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp11.html#T20" title="VECTSP11:th.20">VECTSP11:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:48_3_2_1_2"><i><font color="Green" title="E54">A25</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E9:48_3"><i><font color="Green" title="E52">A23</font></i></a>, <a class="txt" href="#E1:48_3_2_1_2"><i><font color="Green" title="E53">A24</font></i></a>, <a class="txt" href="#E6:48_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="#E1:48_3"><i><font color="Green" title="E47">A18</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="vectsp11.html#T19" title="VECTSP11:th.19">VECTSP11:19</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:48_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E20:48"/><i><font color="Green" title="E47">A26</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E18:48"><i><font color="Green" title="E45">A16</font></i></a>, <a class="txt" href="#E19:48"><i><font color="Green" title="E46">A17</font></i></a>);<br/></i>
<a NAME="E21:48"/><i><font color="Green" title="E48">A27</font></i>: 
 <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">MIN</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c11">D</font> =  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font>;<br/>
<b>assume </b><a NAME="E1:48_4"/><i><font color="Green" title="E49">A28</font></i>: 
 <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">MIN</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:48_4"/>
 <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_4_1"/>
 <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:48_4_1"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="matrixj2.html#T15" target="_self" title="MATRIXJ2:th.15">Th15</a></i>;<br/>
<a NAME="E3:48_4_1"/><b>then </b>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c2">V1</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></i>;<br/>
<a NAME="E4:48_4_1"/><b>then </b>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="ref" href="vectsp_9.html#T33" title="VECTSP_9:th.33">VECTSP_9:33</a></i>;<br/>
<b>hence </b><a NAME="E5:48_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:48"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="matrlin2.html#T21" title="MATRLIN2:th.21">MATRLIN2:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:48_4"/><b>then </b>
 <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E4:48_4"/><b>then </b>
 <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">MIN</font>
 
<b>by </b><i><a class="txt" href="#E1:48_4"><i><font color="Green" title="E49">A28</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E50">A29</font></i>: <font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <span class="p2">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:48_4"><i><font color="Green" title="E49">A28</font></i></a>, <a class="txt" href="#E20:48"><i><font color="Green" title="E47">A26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c2">V1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixj2.html#D5" target="_self" title="MATRIXJ2:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="grcat_1.html#D12" title="GRCAT_1:def.12">GRCAT_1:def 12</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E6:48_4"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">MIN</font> )
 
<b>by </b><i><a class="txt" href="#E1:48_4"><i><font color="Green" title="E49">A28</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:48_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">MIN</font>
 
;<br/>
<a NAME="E8:48_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">i</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">b1</font> is    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c2">V1</font> )
 
<b>by </b><i><a class="txt" href="#E4:48"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="#E13:48"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="#E14:48"><i><font color="Green" title="E43">A14</font></i></a>, <a class="ref" href="matrixj1.html#T10" title="MATRIXJ1:th.10">MATRIXJ1:10</a>, <a class="ref" href="matrlin.html#D4" title="MATRLIN:def.4">MATRLIN:def 4</a></i>;<br/>
<a NAME="E9:48_4"/><b>then </b>
( <font color="Maroon" title="c3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c3">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="matrixj1.html#K10" title="MATRIXJ1:func.10">Len</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">b1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">b1</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V1</font> )
 
<b>by </b><i><a class="txt" href="#E3:48"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="#E17:48"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a>, <a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<b>hence </b><a NAME="E10:48_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:48_4"><i><font color="Green" title="E50">A29</font></i></a>, <a class="ref" href="vectsp_7.html#T3" title="VECTSP_7:th.3">VECTSP_7:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E22:48"/>
<font color="Maroon" title="c7">F</font> <a href="vectsp11.html#K1" title="VECTSP11:func.1">|^</a> <font color="Maroon" title="c8">MIN</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">ZeroMap</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c2">V1</font>
 
<b>by </b><i><a class="txt" href="#E1:48"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="#E5:48"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="#E9:48"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="matrixj2.html#T25" target="_self" title="MATRIXJ2:th.25">Th25</a></i>;<br/>
<a NAME="E23:48"/><b>then </b>
 <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">MIN</font>
 
<b>by </b><i><a class="ref" href="matrixj2.html#D5" target="_self" title="MATRIXJ2:def.5">Def5</a></i>;<br/>
<a NAME="E24:48"/><b>then </b>
 <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">MIN</font>
 
<b>by </b><i><a class="txt" href="#E21:48"><i><font color="Green" title="E48">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E25:48"/>
(  ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">J</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrixj2.html#NK10" title="MATRIXJ2:NK.10">deg</a> <font color="Maroon" title="c7">F</font> ) )
 <b>by </b><i><a class="txt" href="#E9:48"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="#E11:48"><i><font color="Green" title="E41">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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