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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>, <font color="Olive" title="b2">V2</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="b3">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="b4">b2</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b2">V2</font><br/> for <font color="Olive" title="b5">L</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">b1</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="Olive" title="b4">b2</font> holds <br/>for <font color="Olive" title="b6">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b2">V2</font>  st ( for <font color="Olive" title="b7">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <font color="Olive" title="b7">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">b1</font> or <font color="Olive" title="b6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">i</font></span>)</span> or ( <font color="Olive" title="b7">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">b1</font> &amp; <font color="Olive" title="b6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b7">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Olive" title="b5">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b6">F</font>,<font color="Olive" title="b3">b1</font>,<font color="Olive" title="b4">b2</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="b7">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></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">V1</font>, <font color="Olive" title="b2">V2</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="b3">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="b4">b2</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b2">V2</font><br/> for <font color="Olive" title="b5">L</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">b1</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="Olive" title="b4">b2</font> holds <br/>for <font color="Olive" title="b6">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b2">V2</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b7">i</font> where <font color="Olive" title="b7">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b7">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">b1</font> &amp; <font color="Olive" title="b6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">i</font></span>)</span> ) </span>}</span>  &amp; ( for <font color="Olive" title="b7">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <font color="Olive" title="b7">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">b1</font> or <font color="Olive" title="b6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">i</font></span>)</span> or ( <font color="Olive" title="b7">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">b1</font> &amp; <font color="Olive" title="b6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b7">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b7">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Olive" title="b5">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b6">F</font>,<font color="Olive" title="b3">b1</font>,<font color="Olive" title="b4">b2</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="b7">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>;<br/>
<a NAME="E1:50"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">V1</font>, <font color="Maroon" title="c3">V2</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">b2</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V2</font><br/> for <font color="Olive" title="b3">L</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">b1</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="Olive" title="b2">b2</font> holds <br/>for <font color="Olive" title="b4">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>  st  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b5">i</font> where <font color="Olive" title="b5">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="b1">b1</font> &amp; <font color="Olive" title="b4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span> ) </span>}</span>  &amp; ( for <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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="b1">b1</font> or <font color="Olive" title="b4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span> or ( <font color="Olive" title="b5">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">b1</font> &amp; <font color="Olive" title="b4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b5">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Olive" title="b3">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b4">F</font>,<font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</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="b5">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></span><br/><b>let </b><font color="Maroon" title="c4">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">b2</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V2</font><br/> for <font color="Olive" title="b2">L</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</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="Olive" title="b1">b2</font> holds <br/>for <font color="Olive" title="b3">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>  st  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">i</font> where <font color="Olive" title="b4">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="Maroon" title="c4">b1</font> &amp; <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span> ) </span>}</span>  &amp; ( for <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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="Maroon" title="c4">b1</font> or <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span> or ( <font color="Olive" title="b4">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b4">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Olive" title="b2">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b3">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Olive" title="b1">b2</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="b4">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></span><br/><b>let </b><font color="Maroon" title="c5">b2</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V2</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">L</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</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="c5">b2</font> holds <br/>for <font color="Olive" title="b2">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>  st  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">i</font> where <font color="Olive" title="b3">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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">b1</font> &amp; <font color="Olive" title="b2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span> ) </span>}</span>  &amp; ( for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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">b1</font> or <font color="Olive" title="b2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span> or ( <font color="Olive" title="b3">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Olive" title="b2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></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="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b3">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Olive" title="b1">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b2">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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></span><br/><b>let </b><font color="Maroon" title="c6">L</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</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="c5">b2</font> implies for <font color="Olive" title="b1">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>  st  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">i</font> where <font color="Olive" title="b2">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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">b1</font> &amp; <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> ) </span>}</span>  &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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">b1</font> or <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> or ( <font color="Olive" title="b2">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></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="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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> )</span><br/>









<b>assume </b><a NAME="E1:50_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</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="c5">b2</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">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>  st  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">i</font> where <font color="Olive" title="b2">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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">b1</font> &amp; <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> ) </span>}</span>  &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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">b1</font> or <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> or ( <font color="Olive" title="b2">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></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="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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></span><br/>

<b>let </b><font color="Maroon" title="c7">F</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span>  &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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">b1</font> or <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> or ( <font color="Olive" title="b1">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></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="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) implies  ex <font color="Olive" title="b1">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:50_1"/><i><font color="Green" title="E36">A2</font></i>: 
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span> 
 <b>and </b><br/><a NAME="E3:50_1"/><i><font color="Green" title="E37">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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">b1</font> or <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> or ( <font color="Olive" title="b1">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></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="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )
 ; <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">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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></span><br/>

<b>set </b><font color="Maroon" title="c8">Lb</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font>;<br/>
<a NAME="E4:50_1"/><i><font color="Green" title="E38">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:50_1_1"/><b>then </b><i><font color="Green" title="E39">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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="c4">b1</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">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="c4">b1</font></span>)</span> )
 
<b>by </b><i><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></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_1_1"/>
( <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> or ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></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="c5">b2</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="#E3:50_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="#E2:50_1_1"><i><font color="Green" title="E39">A5</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_1_1_1"/>
<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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:50_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E2:50_1_1"><i><font color="Green" title="E39">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E3:50_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:50_1"><i><font color="Green" title="E36">A2</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_1_1_2"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></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="c5">b2</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:50_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E3:50_1_1_1_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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c9">J</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c1">K</font> <b>by </b><i><a class="ref" href="matrixj2.html#T10" target="_self" title="MATRIXJ2:th.10">Th10</a></i>;<br/>
<a NAME="E6:50_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">J</font> holds <br/> not <font color="Maroon" title="c9">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive" title="b1">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">J</font> = <font color="Maroon" title="c9">J</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c1">K</font> <b>by </b><i><a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">J</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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="c10">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></span><br/>

<a NAME="E8:50_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <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="txt" href="#E4:50_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E9:50_1"/><b>then </b>
(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c10">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="matrixj1.html#T22" title="MATRIXJ1:th.22">MATRIXJ1:22</a></i>;<br/>
<b>hence </b><a NAME="E10:50_1"/>
 <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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="c10">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>
 ; <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="E2:50"/><i><font color="Green" title="E36">A6</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>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</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="50_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">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>1</sub></font>[<font color="Maroon" title="c2">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:50_2"/><i><font color="Green" title="E37">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">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>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<b>set </b><font color="Maroon" title="c3">n1</font> = <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>let </b><font color="Maroon" title="c4">V1</font>, <font color="Maroon" title="c5">V2</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="c4">V1</font><br/> for <font color="Olive" title="b2">b2</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c5">V2</font><br/> for <font color="Olive" title="b3">L</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">b1</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="Olive" title="b2">b2</font> holds <br/>for <font color="Olive" title="b4">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c4">V1</font>,<font color="Maroon" title="c5">V2</font>  st <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b5">i</font> where <font color="Olive" title="b5">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="b1">b1</font> &amp; <font color="Olive" title="b4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span> ) </span>}</span>  &amp; ( for <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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="b1">b1</font> or <font color="Olive" title="b4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span> or ( <font color="Olive" title="b5">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">b1</font> &amp; <font color="Olive" title="b4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b5">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Olive" title="b3">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b4">F</font>,<font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</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="b5">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></span><br/><b>let </b><font color="Maroon" title="c6">b1</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">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">b2</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c5">V2</font><br/> for <font color="Olive" title="b2">L</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</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="Olive" title="b1">b2</font> holds <br/>for <font color="Olive" title="b3">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c4">V1</font>,<font color="Maroon" title="c5">V2</font>  st <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">i</font> where <font color="Olive" title="b4">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="Maroon" title="c6">b1</font> &amp; <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span> ) </span>}</span>  &amp; ( for <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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="Maroon" title="c6">b1</font> or <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span> or ( <font color="Olive" title="b4">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b4">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Olive" title="b2">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b3">F</font>,<font color="Maroon" title="c6">b1</font>,<font color="Olive" title="b1">b2</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="b4">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></span><br/><b>let </b><font color="Maroon" title="c7">b2</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c5">V2</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">L</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</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="c7">b2</font> holds <br/>for <font color="Olive" title="b2">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c4">V1</font>,<font color="Maroon" title="c5">V2</font>  st <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">i</font> where <font color="Olive" title="b3">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c6">b1</font> &amp; <font color="Olive" title="b2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span> ) </span>}</span>  &amp; ( for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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="c6">b1</font> or <font color="Olive" title="b2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span> or ( <font color="Olive" title="b3">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Olive" title="b2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></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="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b3">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Olive" title="b1">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b2">F</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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></span><br/><b>let </b><font color="Maroon" title="c8">L</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</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="c7">b2</font> implies for <font color="Olive" title="b1">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c4">V1</font>,<font color="Maroon" title="c5">V2</font>  st <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">i</font> where <font color="Olive" title="b2">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c6">b1</font> &amp; <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> ) </span>}</span>  &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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="c6">b1</font> or <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> or ( <font color="Olive" title="b2">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></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="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c8">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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> )</span><br/>









<b>assume </b><a NAME="E2:50_2"/><i><font color="Green" title="E38">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</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="c7">b2</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">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c4">V1</font>,<font color="Maroon" title="c5">V2</font>  st <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">i</font> where <font color="Olive" title="b2">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c6">b1</font> &amp; <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> ) </span>}</span>  &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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="c6">b1</font> or <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> or ( <font color="Olive" title="b2">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></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="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c8">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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></span><br/>

<b>set </b><font color="Maroon" title="c9">Lb</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font>;<br/>
<b>let </b><font color="Maroon" title="c10">F</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c4">V1</font>,<font color="Maroon" title="c5">V2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c6">b1</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span>  &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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="c6">b1</font> or <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> or ( <font color="Olive" title="b1">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></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="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) implies  ex <font color="Olive" title="b1">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c8">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:50_2"/><i><font color="Green" title="E39">A9</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c6">b1</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:50_2"/><i><font color="Green" title="E40">A10</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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="c6">b1</font> or <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> or ( <font color="Olive" title="b1">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></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="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )
 ; <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">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c8">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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></span><br/>

<b>set </b><font color="Maroon" title="c11">FF</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c6">b1</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">FF</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c6">b1</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="#E3:50_2"><i><font color="Green" title="E39">A9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:50_2"/><i><font color="Green" title="E41">A11</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">FF</font>
 <b>by </b><i><a class="txt" href="#E3:50_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">ii</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:50_2"/><i><font color="Green" title="E42">A12</font></i>: 
( <font color="Maroon" title="c14">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c14">ii</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="c6">b1</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">ii</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">ii</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E7:50_2"><i><font color="Green" title="E41">A11</font></i></a></i>;<br/>
<a NAME="E10:50_2"/><i><font color="Green" title="E43">A13</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">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="c6">b1</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">b2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</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>  )
 
<b>by </b><i><a class="txt" href="#E2:50_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="#E9:50_2"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E11:50_2"/><b>then </b><i><font color="Green" title="E44">A14</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</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="c6">b1</font> &amp; not <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">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>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a>, <a class="ref" href="finseq_3.html#T9" title="FINSEQ_3:th.9">FINSEQ_3:9</a></i>;<br/>
<a NAME="E12:50_2"/><b>then </b>
<font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:50_2"><i><font color="Green" title="E40">A10</font></i></a></i>;<br/>
<a NAME="E13:50_2"/><b>then </b><i><font color="Green" title="E45">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">FF</font>
 
<b>by </b><i><a class="txt" href="#E11:50_2"><i><font color="Green" title="E44">A14</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_2_1"/>
( <font color="Maroon" title="c2">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="c2">n</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1"><b>suppose </b></a><a NAME="E1:50_2_1_1"/><i><font color="Green" title="E46">A16</font></i>: 
<font color="Maroon" title="c2">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">  ex <font color="Olive" title="b1">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c8">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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></span><br/><div class="add"><b>set </b><font color="Maroon" title="c15">J</font> =  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c8">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c16">JJ</font> = <span class="p1"><a href="matrixj1.html#K16" title="MATRIXJ1:func.16">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c8">L</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span>)</span></span><a href="matrixj1.html#K16" title="MATRIXJ1:func.16">*&gt;</a></span> as    <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c8">L</font>,<font color="Maroon" title="c1">K</font> <b>by </b><i><a class="ref" href="matrixj2.html#T11" target="_self" title="MATRIXJ2:th.11">Th11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">x</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="c16">JJ</font> implies  not <font color="Maroon" title="c16">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c17">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/><b>assume </b><a NAME="E1:50_2_1_1_1"/><i><font color="Green" title="E47">A17</font></i>: 
<font color="Maroon" title="c17">x</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="c16">JJ</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c16">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c17">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/><a NAME="E2:50_2_1_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">JJ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default">1</span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E3:50_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c16">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c8">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span> &amp; <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c8">L</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span>)</span> <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="c6">b1</font> )
 <b>by </b><i><a class="txt" href="#E1:50_2_1_1_1"><i><font color="Green" title="E47">A17</font></i></a>, <a class="ref" href="matrixj2.html#D1" target="_self" title="MATRIXJ2:def.1">Def1</a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:50_2_1_1_1"/>
not <font color="Maroon" title="c16">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c17">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="#E10:50_2"><i><font color="Green" title="E43">A13</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/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c17">JJ</font> = <font color="Maroon" title="c16">JJ</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c8">L</font>,<font color="Maroon" title="c1">K</font> <b>by </b><i><a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/><b>take </b>
<font color="Maroon" title="c17">JJ</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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="c17">JJ</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></span><br/><a NAME="E5:50_2_1_1"/><i><font color="Green" title="E47">A18</font></i>: 
 <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c17">JJ</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c8">L</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrixj1.html#T34" title="MATRIXJ1:th.34">MATRIXJ1:34</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">BB</font> =  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c17">JJ</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> as   <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="c6">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c1">K</font> <b>by </b><i><a class="txt" href="#E2:50_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="matrixj1.html#T34" title="MATRIXJ1:th.34">MATRIXJ1:34</a></i>;<br/><b>set </b><font color="Maroon" title="c19">T</font> =  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c18">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font>;<br/><b>reconsider </b><font color="Maroon" title="c20">F'</font> = <font color="Maroon" title="c10">F</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">V1</font>,<font color="Maroon" title="c5">V2</font> ;<br/><a NAME="E8:50_2_1_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">V1</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">V1</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c18">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">V1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E9:50_2_1_1"/><b>then </b><i><font color="Green" title="E48">A19</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c18">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c21">y</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="c6">b1</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c20">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</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="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c18">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font> )</span><br/><b>assume </b><a NAME="E1:50_2_1_1_2"/><i><font color="Green" title="E49">A20</font></i>: 
<font color="Maroon" title="c21">y</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="c6">b1</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="c20">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</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="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c18">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font></span><br/><b>reconsider </b><font color="Maroon" title="c22">j</font> = <font color="Maroon" title="c21">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:50_2_1_1_2"><i><font color="Green" title="E49">A20</font></i></a></i>;<br/><a NAME="E3:50_2_1_1_2"/><i><font color="Green" title="E50">A21</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c18">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c18">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font></span>)</span> &amp; <font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">j</font> )
 <b>by </b><i><a class="txt" href="#E1:50_2_1_1_2"><i><font color="Green" title="E49">A20</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_2_1_1_2_1_1"/>
( <font color="Maroon" title="c22">j</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="c6">b1</font> or <font color="Maroon" title="c22">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:50_2_1_1_2_1_1_1"/><i><font color="Green" title="E51">A22</font></i>: 
<font color="Maroon" title="c22">j</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="c6">b1</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"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c18">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c18">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font> = 
<font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:50_2_1_1"><i><font color="Green" title="E47">A18</font></i></a>, <a class="txt" href="#E1:50_2_1_1_2"><i><font color="Green" title="E49">A20</font></i></a>, <a class="txt" href="#E3:50_2_1_1_2"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="matrixj2.html#T23" target="_self" title="MATRIXJ2:th.23">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font>
<b>by </b><i><a class="txt" href="#E11:50_2"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="#E4:50_2"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="#E3:50_2_1_1_2"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="#E1:50_2_1_1_2_1_1_1"><i><font color="Green" title="E51">A22</font></i></a></i>
;<br/> <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="50_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:50_2_1_1_2_1_1_2"/><i><font color="Green" title="E51">A23</font></i>: 
<font color="Maroon" title="c22">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</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"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c18">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font></span><br/><div class="add"><a NAME="E2:50_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E52">A24</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c18">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">j</font></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="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">j</font> <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="#E5:50_2_1_1"><i><font color="Green" title="E47">A18</font></i></a>, <a class="txt" href="#E1:50_2_1_1_2"><i><font color="Green" title="E49">A20</font></i></a>, <a class="txt" href="#E3:50_2_1_1_2"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="matrixj2.html#T23" target="_self" title="MATRIXJ2:th.23">Th23</a></i>;<br/><a NAME="E3:50_2_1_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon" title="c12">FF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E3:50_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="#E1:50_2_1_1"><i><font color="Green" title="E46">A16</font></i></a>, <a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></i>;<br/><a NAME="E4:50_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">FF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <b>by </b><i><a class="txt" href="#E13:50_2"><i><font color="Green" title="E45">A15</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:50_2_1_1_2_1_1_2"/><b>then </b>
not <font color="Maroon" title="c22">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">FF</font>
 <b>by </b><i><a class="txt" href="#E1:50_2_1_1_2_1_1_2"><i><font color="Green" title="E51">A23</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E6:50_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:50_2_1_1_2"><i><font color="Green" title="E49">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E7:50_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c18">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font>
 <b>by </b><i><a class="txt" href="#E1:50_2_1_1_2"><i><font color="Green" title="E49">A20</font></i></a>, <a class="txt" href="#E3:50_2_1_1_2"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="#E2:50_2_1_1_2_1_1_2"><i><font color="Green" title="E52">A24</font></i></a>, <a class="txt" href="#E4:50_2"><i><font color="Green" title="E40">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/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:50_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</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="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c18">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">y</font>
 ; <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="E11:50_2_1_1"/><b>then </b>
<font color="Maroon" title="c10">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="c18">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font>
 <b>by </b><i><a class="txt" href="#E11:50_2"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="#E9:50_2_1_1"><i><font color="Green" title="E48">A19</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a>, <a class="ref" href="matrlin.html#T26" title="MATRLIN:th.26">MATRLIN:26</a></i>;<br/><b>hence </b><a NAME="E12:50_2_1_1"/>
 <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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="c17">JJ</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>by </b><i><a class="ref" href="matrlin2.html#T36" title="MATRLIN2:th.36">MATRLIN2:36</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="50_2_1_2"><b>suppose </b></a><a NAME="E1:50_2_1_2"/><i><font color="Green" title="E46">A25</font></i>: 
<font color="Maroon" title="c2">n</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> 
 ; <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">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c8">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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></span><br/><div class="add"><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> <font color="Maroon" title="c12">FF</font> &amp; $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font> );<br/><a NAME="E2:50_2_1_2"/><i><font color="Green" title="E47">A26</font></i>: 
for <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>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Olive" title="b1">k</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> <font color="Maroon" title="c6">b1</font>
 ;<br/><a NAME="E3:50_2_1_2"/><i><font color="Green" title="E48">A27</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>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_2_1_2_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">FF</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="#E3:50_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="#E1:50_2_1_2"><i><font color="Green" title="E46">A25</font></i></a></i>;<br/>
<a NAME="E2:50_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c12">FF</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 
<b>by </b><i><a class="ref" href="card_2.html#T60" title="CARD_2:th.60">CARD_2:60</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:50_2_1_2_1"/><i><font color="Green" title="E49">A28</font></i>: 
( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">FF</font> &amp; <font color="Maroon" title="c15">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font> )
 <b>by </b><i><a class="txt" href="#E13:50_2"><i><font color="Green" title="E45">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#T41" title="ZFMISC_1:th.41">ZFMISC_1:41</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:50_2_1_2_1"/><i><font color="Green" title="E50">A29</font></i>: 
( <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font> &amp; <font color="Maroon" title="c16">j</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="c6">b1</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:50_2_1_2_1"><i><font color="Green" title="E49">A28</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c16">j</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="c16">j</font>]</span><br/>

<a NAME="E7:50_2_1_2_1"/>
<font color="Maroon" title="c16">j</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> <font color="Maroon" title="c6">b1</font>
 
<b>by </b><i><a class="txt" href="#E6:50_2_1_2_1"><i><font color="Green" title="E50">A29</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E8:50_2_1_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">j</font>]
 <b>by </b><i><a class="txt" href="#E4:50_2_1_2_1"><i><font color="Green" title="E49">A28</font></i></a>, <a class="txt" href="#E6:50_2_1_2_1"><i><font color="Green" title="E50">A29</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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>consider </b><font color="Maroon" title="c15">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:50_2_1_2"/><i><font color="Green" title="E49">A30</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">m</font>]
 <b>and </b><br/><a NAME="E6:50_2_1_2"/><i><font color="Green" title="E50">A31</font></i>: 
for <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>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="#E2:50_2_1_2"><i><font color="Green" title="E47">A26</font></i></a>, <a class="txt" href="#E3:50_2_1_2"><i><font color="Green" title="E48">A27</font></i></a>);<br/></i><b>set </b><font color="Maroon" title="c16">b1m</font> = <font color="Maroon" title="c6">b1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">m</font>;<br/><b>set </b><font color="Maroon" title="c17">b2m</font> = <font color="Maroon" title="c7">b2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">m</font>;<br/><b>set </b><font color="Maroon" title="c18">b1'm</font> = <font color="Maroon" title="c6">b1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">m</font>;<br/><b>set </b><font color="Maroon" title="c19">b2'm</font> = <font color="Maroon" title="c7">b2</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">m</font>;<br/><a NAME="E7:50_2_1_2"/><i><font color="Green" title="E51">A32</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="c6">b1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m</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="c7">b2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m</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="c6">b1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">m</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="c7">b2</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">m</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c15">m</font> )
 <b>by </b><i><a class="txt" href="#E2:50_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="#E5:50_2_1_2"><i><font color="Green" title="E49">A30</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">Rb1</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">m</font></span>)</span>, <font color="Maroon" title="c21">Rb1'</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">m</font></span>)</span> as  <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="c4">V1</font> <b>by </b><i><a class="ref" href="matrlin2.html#T22" title="MATRLIN2:th.22">MATRLIN2:22</a>, <a class="ref" href="matrlin2.html#T23" title="MATRLIN2:th.23">MATRLIN2:23</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">Rb2</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">m</font></span>)</span>, <font color="Maroon" title="c23">Rb2'</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">m</font></span>)</span> as  <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="c5">V2</font> <b>by </b><i><a class="ref" href="matrlin2.html#T22" title="MATRLIN2:th.22">MATRLIN2:22</a>, <a class="ref" href="matrlin2.html#T23" title="MATRLIN2:th.23">MATRLIN2:23</a></i>;<br/><b>set </b><font color="Maroon" title="c24">Lb1</font> =  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</font>;<br/><b>set </b><font color="Maroon" title="c25">Lb2</font> =  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c22">Rb2</font>;<br/><b>set </b><font color="Maroon" title="c26">Lb1'</font> =  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c21">Rb1'</font>;<br/><b>set </b><font color="Maroon" title="c27">Lb2'</font> =  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c23">Rb2'</font>;<br/><b>reconsider </b><font color="Maroon" title="c28">b1m</font> = <font color="Maroon" title="c6">b1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">m</font> as    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</font> <b>by </b><i><a class="ref" href="matrlin2.html#T22" title="MATRLIN2:th.22">MATRLIN2:22</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">b1'm</font> = <font color="Maroon" title="c6">b1</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">m</font> as    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c21">Rb1'</font> <b>by </b><i><a class="ref" href="matrlin2.html#T23" title="MATRLIN2:th.23">MATRLIN2:23</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">b2m</font> = <font color="Maroon" title="c7">b2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">m</font> as    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c22">Rb2</font> <b>by </b><i><a class="ref" href="matrlin2.html#T22" title="MATRLIN2:th.22">MATRLIN2:22</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c31">b2'm</font> = <font color="Maroon" title="c7">b2</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">m</font> as    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c23">Rb2'</font> <b>by </b><i><a class="ref" href="matrlin2.html#T23" title="MATRLIN2:th.23">MATRLIN2:23</a></i>;<br/><a NAME="E14:50_2_1_2"/><i><font color="Green" title="E52">A33</font></i>: 
( <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">b1m</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c29">b1'm</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">b2m</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c31">b2'm</font> )
 <b>by </b><i><a class="ref" href="rfinseq.html#T21" title="RFINSEQ:th.21">RFINSEQ:21</a></i>;<br/><a NAME="E15:50_2_1_2"/><b>then </b><i><font color="Green" title="E53">A34</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">b1m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">b1'm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c31">b2'm</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">b1m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c30">b2m</font> )
 <b>by </b><i><a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><b>set </b><font color="Maroon" title="c32">FRb1</font> = <font color="Maroon" title="c10">F</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c20">Rb1</font>;<br/><a NAME="E16:50_2_1_2"/><i><font color="Green" title="E54">A35</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c20">Rb1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp11.html#T42" title="VECTSP11:th.42">VECTSP11:42</a></i>;<br/><a NAME="E17:50_2_1_2"/><i><font color="Green" title="E55">A36</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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="c28">b1m</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> or ( <font color="Olive" title="b1">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">b1m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></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="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c33">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"> (  not <font color="Maroon" title="c33">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="c28">b1m</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> or ( <font color="Maroon" title="c33">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">b1m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></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="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E1:50_2_1_2_2"/><i><font color="Green" title="E56">A37</font></i>: 
<font color="Maroon" title="c33">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="c28">b1m</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="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> or ( <font color="Maroon" title="c33">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">b1m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></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="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )</span><br/>

<a NAME="E2:50_2_1_2_2"/><i><font color="Green" title="E57">A38</font></i>: 
( <font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">b2m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">b1m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> &amp; <font color="Maroon" title="c28">b1m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font> &amp; <font color="Maroon" title="c30">b2m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E10:50_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="#E1:50_2_1_2_2"><i><font color="Green" title="E56">A37</font></i></a>, <a class="txt" href="#E14:50_2_1_2"><i><font color="Green" title="E52">A33</font></i></a>, <a class="txt" href="#E15:50_2_1_2"><i><font color="Green" title="E53">A34</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="c34">i1</font> = <font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_2_1_2_2_1"/>
( <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> or ( <font color="Maroon" title="c33">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></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="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="#E1:50_2_1_2_2"><i><font color="Green" title="E56">A37</font></i></a>, <a class="txt" href="#E4:50_2"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="#E15:50_2_1_2"><i><font color="Green" title="E53">A34</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:50_2_1_2_2_1_1"/>
<font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span>
 ; <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="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> or ( <font color="Maroon" title="c33">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">b1m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></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="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )</span><br/><div class="add"><b>hence </b><a NAME="E2:50_2_1_2_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> or ( <font color="Maroon" title="c33">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">b1m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></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="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="#E2:50_2_1_2_2"><i><font color="Green" title="E57">A38</font></i></a>, <a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</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="50_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:50_2_1_2_2_1_2"/><i><font color="Green" title="E58">A39</font></i>: 
( <font color="Maroon" title="c33">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></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="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 ; <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="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> or ( <font color="Maroon" title="c33">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">b1m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></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="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c35">rngb2</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">b2</font> as    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c5">V2</font> <b>by </b><i><a class="ref" href="matrlin.html#D4" title="MATRLIN:def.4">MATRLIN:def 4</a></i>;<br/><a NAME="E3:50_2_1_2_2_1_2"/><i><font color="Green" title="E59">A40</font></i>: 
<font color="Maroon" title="c33">i</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="c15">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_2_2_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_2_1_2_2_1_2_1"/>
<font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">m</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="E2:50_2_1_2_2_1_2_1"/><b>then </b>
( <font color="Maroon" title="c33">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">m</font> &amp; <font color="Maroon" title="c33">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font> &amp;  ex <font color="Olive" title="b1">ii</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ii</font> &amp; <font color="Olive" title="b1">ii</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="c6">b1</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">ii</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">ii</font></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="#E5:50_2_1_2"><i><font color="Green" title="E49">A30</font></i></a>, <a class="txt" href="#E1:50_2_1_2_2"><i><font color="Green" title="E56">A37</font></i></a>, <a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:50_2_1_2_2_1_2_1"/><b>then </b>
( <font color="Maroon" title="c33">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">m</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E4:50_2_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">V2</font></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/>
<a NAME="E5:50_2_1_2_2_1_2_1"/><b>then </b>
( <font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">V2</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">i</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> <font color="Maroon" title="c35">rngb2</font> &amp; <font color="Maroon" title="c35">rngb2</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> )
 
<b>by </b><i><a class="txt" href="#E10:50_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="#E1:50_2_1_2_2_1_2"><i><font color="Green" title="E58">A39</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="rlvect_1.html#T21" title="RLVECT_1:th.21">RLVECT_1:21</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="E6:50_2_1_2_2_1_2_1"/>
contradiction
 <b>by </b><i><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="E4:50_2_1_2_2_1_2"/><i><font color="Green" title="E60">A41</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E5:50_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c33">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">b1m</font>
 <b>by </b><i><a class="txt" href="#E3:50_2_1_2_2_1_2"><i><font color="Green" title="E59">A40</font></i></a>, <a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:50_2_1_2_2_1_2"/><b>then </b>
( <font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">b2m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c30">b2m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:50_2_1_2_2"><i><font color="Green" title="E57">A38</font></i></a>, <a class="txt" href="#E10:50_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="#E14:50_2_1_2"><i><font color="Green" title="E52">A33</font></i></a>, <a class="txt" href="#E15:50_2_1_2"><i><font color="Green" title="E53">A34</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a>, <a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</a></i>;<br/><b>hence </b><a NAME="E7:50_2_1_2_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> or ( <font color="Maroon" title="c33">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">b1m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c33">i</font></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="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="#E2:50_2_1_2_2"><i><font color="Green" title="E57">A38</font></i></a>, <a class="txt" href="#E1:50_2_1_2_2_1_2"><i><font color="Green" title="E58">A39</font></i></a>, <a class="txt" href="#E4:50_2_1_2_2_1_2"><i><font color="Green" title="E60">A41</font></i></a>, <a class="txt" href="#E3:50_2_1_2_2_1_2"><i><font color="Green" title="E59">A40</font></i></a>, <a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="vectsp_4.html#T21" title="VECTSP_4:th.21">VECTSP_4:21</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></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E18:50_2_1_2"/>
<font color="Maroon" title="c10">F</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c20">Rb1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c22">Rb2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c33">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c33">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c20">Rb1</font> or <font color="Maroon" title="c33">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c22">Rb2</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:50_2_1_2_3"/><i><font color="Green" title="E56">A42</font></i>: 
<font color="Maroon" title="c33">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c20">Rb1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c33">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c22">Rb2</font></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c34">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:50_2_1_2_3"/><i><font color="Green" title="E57">A43</font></i>: 
( <font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">F</font> &amp; <font color="Maroon" title="c34">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Rb1</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">y</font> )
 <b>by </b><i><a class="txt" href="#E1:50_2_1_2_3"><i><font color="Green" title="E56">A42</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c35">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:50_2_1_2_3"/><i><font color="Green" title="E58">A44</font></i>: 
( <font color="Maroon" title="c35">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="c28">b1m</font> &amp; <font color="Maroon" title="c28">b1m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">x</font> )
 <b>by </b><i><a class="txt" href="#E3:50_2_1_2_3"><i><font color="Green" title="E57">A43</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c36">i</font> = <font color="Maroon" title="c35">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E5:50_2_1_2_3"><i><font color="Green" title="E58">A44</font></i></a></i>;<br/>
<a NAME="E7:50_2_1_2_3"/>
<font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">b1m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">i</font>
 
<b>by </b><i><a class="txt" href="#E5:50_2_1_2_3"><i><font color="Green" title="E58">A44</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E8:50_2_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">y</font>
 
<b>by </b><i><a class="txt" href="#E3:50_2_1_2_3"><i><font color="Green" title="E57">A43</font></i></a>, <a class="txt" href="#E5:50_2_1_2_3"><i><font color="Green" title="E58">A44</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E9:50_2_1_2_3"/><b>then </b>
( <font color="Maroon" title="c33">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">i</font></span>)</span> or <font color="Maroon" title="c33">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">i</font></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="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c36">i</font> <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="#E5:50_2_1_2_3"><i><font color="Green" title="E58">A44</font></i></a>, <a class="txt" href="#E17:50_2_1_2"><i><font color="Green" title="E55">A36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:50_2_1_2_3"/>
<font color="Maroon" title="c33">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c22">Rb2</font></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="E19:50_2_1_2"/><b>then </b>
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <font color="Maroon" title="c20">Rb1</font></span>)</span> is    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c22">Rb2</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T20" title="VECTSP_9:th.20">VECTSP_9:20</a></i>;<br/><a NAME="E20:50_2_1_2"/><b>then </b>
( <font color="Maroon" title="c10">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c22">Rb2</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E16:50_2_1_2"><i><font color="Green" title="E54">A35</font></i></a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a>, <a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c33">FL</font> = <font color="Maroon" title="c10">F</font> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</font></span>)</span> as   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c20">Rb1</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c22">Rb2</font></span>)</span> <b>by </b><i><a class="txt" href="matrixj2.html#E36"><i><font color="Green" title="E33">Lm4</font></i></a>, <a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></i>;<br/><a NAME="E22:50_2_1_2"/><i><font color="Green" title="E56">A45</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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="c28">b1m</font> or <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> or ( <font color="Olive" title="b1">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">b1m</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></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="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="#E17:50_2_1_2"><i><font color="Green" title="E55">A36</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c34">FF1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c28">b1m</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span> ;<br/><a NAME="E23:50_2_1_2"/><i><font color="Green" title="E57">A46</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c28">b1m</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">FF</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c35">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c28">b1m</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span>  or <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">FF</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:50_2_1_2_4"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c28">b1m</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">FF</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span></span><br/>

<b>then consider </b><font color="Maroon" title="c36">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:50_2_1_2_4"/><i><font color="Green" title="E58">A47</font></i>: 
( <font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">j</font> &amp; <font color="Maroon" title="c36">j</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="c28">b1m</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font></span>)</span> )
 ;<br/>
<a NAME="E4:50_2_1_2_4"/>
( <font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">b1m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> &amp; <font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">b2m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> &amp; <font color="Maroon" title="c28">b1m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> &amp; <font color="Maroon" title="c30">b2m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font></span>)</span> &amp; <font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> )
 
<b>by </b><i><a class="txt" href="#E10:50_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="#E14:50_2_1_2"><i><font color="Green" title="E52">A33</font></i></a>, <a class="txt" href="#E15:50_2_1_2"><i><font color="Green" title="E53">A34</font></i></a>, <a class="txt" href="#E3:50_2_1_2_4"><i><font color="Green" title="E58">A47</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E5:50_2_1_2_4"/><b>then </b>
<font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:50_2_1_2_4"><i><font color="Green" title="E58">A47</font></i></a>, <a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</a></i>;<br/>
<a NAME="E6:50_2_1_2_4"/><b>then </b><i><font color="Green" title="E59">A48</font></i>: 
<font color="Maroon" title="c36">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">FF</font>
 
<b>by </b><i><a class="txt" href="#E3:50_2_1_2_4"><i><font color="Green" title="E58">A47</font></i></a>, <a class="txt" href="#E15:50_2_1_2"><i><font color="Green" title="E53">A34</font></i></a></i>;<br/>
<a NAME="E7:50_2_1_2_4"/>
<font color="Maroon" title="c36">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 
<b>by </b><i><a class="txt" href="#E3:50_2_1_2_4"><i><font color="Green" title="E58">A47</font></i></a>, <a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E8:50_2_1_2_4"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">FF</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <b>by </b><i><a class="txt" href="#E3:50_2_1_2_4"><i><font color="Green" title="E58">A47</font></i></a>, <a class="txt" href="#E6:50_2_1_2_4"><i><font color="Green" title="E59">A48</font></i></a>, <a class="txt" href="#E5:50_2_1_2"><i><font color="Green" title="E49">A30</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</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="E24:50_2_1_2"/>
<font color="Maroon" title="c12">FF</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c28">b1m</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c35">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">FF</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> or <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c28">b1m</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:50_2_1_2_5"/><i><font color="Green" title="E58">A49</font></i>: 
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">FF</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c28">b1m</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span> </span><br/>

<a NAME="E2:50_2_1_2_5"/><i><font color="Green" title="E59">A50</font></i>: 
( <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">FF</font> &amp; <font color="Maroon" title="c35">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font> )
 
<b>by </b><i><a class="txt" href="#E1:50_2_1_2_5"><i><font color="Green" title="E58">A49</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c36">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:50_2_1_2_5"/><i><font color="Green" title="E60">A51</font></i>: 
( <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">x</font> &amp; <font color="Maroon" title="c36">j</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="c6">b1</font> &amp; <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font></span>)</span> )
 ;<br/>
<a NAME="E5:50_2_1_2_5"/>
<font color="Maroon" title="c36">j</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> <font color="Maroon" title="c6">b1</font>
 
<b>by </b><i><a class="txt" href="#E4:50_2_1_2_5"><i><font color="Green" title="E60">A51</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:50_2_1_2_5"/><b>then </b>
<font color="Maroon" title="c36">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font>
 
<b>by </b><i><a class="txt" href="#E2:50_2_1_2_5"><i><font color="Green" title="E59">A50</font></i></a>, <a class="txt" href="#E4:50_2_1_2_5"><i><font color="Green" title="E60">A51</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E7:50_2_1_2_5"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">j</font> &amp; <font color="Maroon" title="c36">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font> )
 
<b>by </b><i><a class="txt" href="#E6:50_2_1_2"><i><font color="Green" title="E50">A31</font></i></a>, <a class="txt" href="#E1:50_2_1_2_5"><i><font color="Green" title="E58">A49</font></i></a>, <a class="txt" href="#E4:50_2_1_2_5"><i><font color="Green" title="E60">A51</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:50_2_1_2_5"/><b>then </b><i><font color="Green" title="E61">A52</font></i>: 
<font color="Maroon" title="c36">j</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="c28">b1m</font>
 
<b>by </b><i><a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:50_2_1_2_5"/><b>then </b>
( <font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">b1m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> &amp; <font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">b2m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> &amp; <font color="Maroon" title="c28">b1m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> &amp; <font color="Maroon" title="c30">b2m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font></span>)</span> &amp; <font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> )
 
<b>by </b><i><a class="txt" href="#E10:50_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="#E14:50_2_1_2"><i><font color="Green" title="E52">A33</font></i></a>, <a class="txt" href="#E15:50_2_1_2"><i><font color="Green" title="E53">A34</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E10:50_2_1_2_5"/><b>then </b>
<font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:50_2_1_2_5"><i><font color="Green" title="E60">A51</font></i></a>, <a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</a></i>;<br/>
<b>hence </b><a NAME="E11:50_2_1_2_5"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c28">b1m</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E4:50_2_1_2_5"><i><font color="Green" title="E60">A51</font></i></a>, <a class="txt" href="#E8:50_2_1_2_5"><i><font color="Green" title="E61">A52</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/>


</div><b>end;</b></div><a NAME="E25:50_2_1_2"/><b>then </b>
( <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c28">b1m</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">FF</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span></span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span> &amp; <font color="Maroon" title="c2">n</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="#E23:50_2_1_2"><i><font color="Green" title="E57">A46</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E26:50_2_1_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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="c28">b1m</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="#E3:50_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="#E13:50_2"><i><font color="Green" title="E45">A15</font></i></a>, <a class="ref" href="stirl2_1.html#T65" title="STIRL2_1:th.65">STIRL2_1:65</a></i>;<br/><b>then consider </b><font color="Maroon" title="c35">J</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c8">L</font>,<font color="Maroon" title="c1">K</font><b> such that </b><br/><a NAME="E28:50_2_1_2"/><i><font color="Green" title="E58">A53</font></i>: 
 <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c33">FL</font>,<font color="Maroon" title="c28">b1m</font>,<font color="Maroon" title="c30">b2m</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="c35">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>by </b><i><a class="txt" href="#E1:50_2"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="txt" href="#E22:50_2_1_2"><i><font color="Green" title="E56">A45</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c36">B</font> =  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c35">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> as   <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="c28">b1m</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">b2m</font>,<font color="Maroon" title="c1">K</font> <b>by </b><i><a class="txt" href="#E28:50_2_1_2"><i><font color="Green" title="E58">A53</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c37">JB</font> =  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c8">L</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c15">m</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c38">JJ</font> = <span class="p1"><a href="matrixj1.html#K16" title="MATRIXJ1:func.16">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c8">L</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span></span><a href="matrixj1.html#K16" title="MATRIXJ1:func.16">*&gt;</a></span> as    <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c8">L</font>,<font color="Maroon" title="c1">K</font> <b>by </b><i><a class="ref" href="matrixj2.html#T11" target="_self" title="MATRIXJ2:th.11">Th11</a></i>;<br/><b>set </b><font color="Maroon" title="c39">JJJ</font> = <font color="Maroon" title="c35">J</font> <a href="matrixj2.html#K6" title="MATRIXJ2:func.6">^</a> <font color="Maroon" title="c38">JJ</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c40">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c40">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c35">J</font> <a href="matrixj2.html#K6" title="MATRIXJ2:func.6">^</a> <font color="Maroon" title="c38">JJ</font></span>)</span> implies  not <span class="p1">(<span class="default"><font color="Maroon" title="c35">J</font> <a href="matrixj2.html#K6" title="MATRIXJ2:func.6">^</a> <font color="Maroon" title="c38">JJ</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/><b>assume </b><a NAME="E1:50_2_1_2_6"/><i><font color="Green" title="E59">A54</font></i>: 
<font color="Maroon" title="c40">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c35">J</font> <a href="matrixj2.html#K6" title="MATRIXJ2:func.6">^</a> <font color="Maroon" title="c38">JJ</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1">(<span class="default"><font color="Maroon" title="c35">J</font> <a href="matrixj2.html#K6" title="MATRIXJ2:func.6">^</a> <font color="Maroon" title="c38">JJ</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/><b>reconsider </b><font color="Maroon" title="c41">i</font> = <font color="Maroon" title="c40">x</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="#E1:50_2_1_2_6"><i><font color="Green" title="E59">A54</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_2_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_2_1_2_6_1"/>
( <font color="Maroon" title="c41">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="c35">J</font> or  ex <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">j</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="c38">JJ</font> &amp; <font color="Maroon" title="c41">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c35">J</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font> ) )
 </span><b>by </b><i><a class="txt" href="#E1:50_2_1_2_6"><i><font color="Green" title="E59">A54</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_2_6_1_1"><b>suppose </b></a><a NAME="E1:50_2_1_2_6_1_1"/>
<font color="Maroon" title="c41">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="c35">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1">(<span class="default"><font color="Maroon" title="c35">J</font> <a href="matrixj2.html#K6" title="MATRIXJ2:func.6">^</a> <font color="Maroon" title="c38">JJ</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/><div class="add"><a NAME="E2:50_2_1_2_6_1_1"/><b>then </b>
( not <font color="Maroon" title="c35">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c41">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c35">J</font> <a href="matrixj2.html#K6" title="MATRIXJ2:func.6">^</a> <font color="Maroon" title="c38">JJ</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c41">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">J</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c41">i</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/><b>hence </b><a NAME="E3:50_2_1_2_6_1_1"/>
not <span class="p1">(<span class="default"><font color="Maroon" title="c35">J</font> <a href="matrixj2.html#K6" title="MATRIXJ2:func.6">^</a> <font color="Maroon" title="c38">JJ</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c40">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <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="50_2_1_2_6_1_2"><b>suppose </b></a><a NAME="E1:50_2_1_2_6_1_2"/>
 ex <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">j</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="c38">JJ</font> &amp; <font color="Maroon" title="c41">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c35">J</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1">(<span class="default"><font color="Maroon" title="c35">J</font> <a href="matrixj2.html#K6" title="MATRIXJ2:func.6">^</a> <font color="Maroon" title="c38">JJ</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c42">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:50_2_1_2_6_1_2"/><i><font color="Green" title="E60">A55</font></i>: 
( <font color="Maroon" title="c42">j</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="c38">JJ</font> &amp; <font color="Maroon" title="c41">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c35">J</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c42">j</font> )
 ;<br/><a NAME="E4:50_2_1_2_6_1_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c38">JJ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E5:50_2_1_2_6_1_2"/><b>then </b>
( <font color="Maroon" title="c42">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c8">L</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c15">m</font> )
 <b>by </b><i><a class="txt" href="#E3:50_2_1_2_6_1_2"><i><font color="Green" title="E60">A55</font></i></a>, <a class="ref" href="matrixj2.html#D1" target="_self" title="MATRIXJ2:def.1">Def1</a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E6:50_2_1_2_6_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c35">J</font> <a href="matrixj2.html#K6" title="MATRIXJ2:func.6">^</a> <font color="Maroon" title="c38">JJ</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c41">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> 1 &amp; <font color="Maroon" title="c38">JJ</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c8">L</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c15">m</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c8">L</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <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>  )
 <b>by </b><i><a class="txt" href="#E5:50_2_1_2"><i><font color="Green" title="E49">A30</font></i></a>, <a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="txt" href="#E3:50_2_1_2_6_1_2"><i><font color="Green" title="E60">A55</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E7:50_2_1_2_6_1_2"/>
not <span class="p1">(<span class="default"><font color="Maroon" title="c35">J</font> <a href="matrixj2.html#K6" title="MATRIXJ2:func.6">^</a> <font color="Maroon" title="c38">JJ</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c40">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <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>then reconsider </b><font color="Maroon" title="c40">JJJ</font> = <font color="Maroon" title="c35">J</font> <a href="matrixj2.html#K6" title="MATRIXJ2:func.6">^</a> <font color="Maroon" title="c38">JJ</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c8">L</font>,<font color="Maroon" title="c1">K</font> <b>by </b><i><a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/><b>take </b>
<font color="Maroon" title="c40">JJJ</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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="c40">JJJ</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></span><br/><b>reconsider </b><font color="Maroon" title="c41">JB</font> =  <a href="matrixj2.html#K2" title="MATRIXJ2:func.2">Jordan_block</a> <font color="Maroon" title="c8">L</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c15">m</font></span>)</span> as   <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="c29">b1'm</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">b2'm</font>,<font color="Maroon" title="c1">K</font> <b>by </b><i><a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a></i>;<br/><a NAME="E34:50_2_1_2"/><i><font color="Green" title="E59">A56</font></i>: 
(  <a href="matrixj1.html#K20" title="MATRIXJ1:func.20">block_diagonal</a> <font color="Maroon" title="c40">JJJ</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1"><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c36">B</font>,<font color="Maroon" title="c41">JB</font></span><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">*&gt;</a></span>,<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="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> <span class="p2"><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c36">B</font>,<font color="Maroon" title="c41">JB</font></span><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">B</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">JB</font></span>)</span> &amp;  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K11" title="MATRIXJ1:func.11">Width</a> <span class="p2"><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c36">B</font>,<font color="Maroon" title="c41">JB</font></span><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c36">B</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c41">JB</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">B</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="c28">b1m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">JB</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="c29">b1'm</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c36">B</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="c28">b1m</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c41">JB</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="c29">b1'm</font> )
 <b>by </b><i><a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="ref" href="matrixj1.html#T16" title="MATRIXJ1:th.16">MATRIXJ1:16</a>, <a class="ref" href="matrixj1.html#T20" title="MATRIXJ1:th.20">MATRIXJ1:20</a>, <a class="ref" href="matrixj1.html#T35" title="MATRIXJ1:th.35">MATRIXJ1:35</a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c42">BB</font> =  <a href="matrixj1.html#K13" title="MATRIXJ1:func.13">block_diagonal</a> <span class="p1"><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c36">B</font>,<font color="Maroon" title="c41">JB</font></span><a href="matrixj1.html#K5" title="MATRIXJ1:func.5">*&gt;</a></span>,<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> as   <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="c6">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c1">K</font> <b>by </b><i><a class="txt" href="#E2:50_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c43">T</font> =  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c42">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font>;<br/><b>reconsider </b><font color="Maroon" title="c44">F'</font> = <font color="Maroon" title="c10">F</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">V1</font>,<font color="Maroon" title="c5">V2</font> ;<br/><a NAME="E37:50_2_1_2"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">V1</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">V1</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c42">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c4">V1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E38:50_2_1_2"/><b>then </b><i><font color="Green" title="E60">A57</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c42">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">b1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c45">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c45">x</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="c6">b1</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c42">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> )</span><br/><b>assume </b><a NAME="E1:50_2_1_2_7"/><i><font color="Green" title="E61">A58</font></i>: 
<font color="Maroon" title="c45">x</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="c6">b1</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="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c42">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font></span><br/><b>reconsider </b><font color="Maroon" title="c46">I</font> = <font color="Maroon" title="c45">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:50_2_1_2_7"><i><font color="Green" title="E61">A58</font></i></a></i>;<br/><a NAME="E3:50_2_1_2_7"/><i><font color="Green" title="E62">A59</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">I</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c42">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c42">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">I</font></span>)</span> &amp; <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font> )
 <b>by </b><i><a class="txt" href="#E1:50_2_1_2_7"><i><font color="Green" title="E61">A58</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_2_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_2_1_2_7_1_1"/>
( <font color="Maroon" title="c46">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="c28">b1m</font> or  ex <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">j</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="c29">b1'm</font> &amp; <font color="Maroon" title="c46">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">b1m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font> ) )
 </span><b>by </b><i><a class="txt" href="#E1:50_2_1_2_7"><i><font color="Green" title="E61">A58</font></i></a>, <a class="txt" href="#E14:50_2_1_2"><i><font color="Green" title="E52">A33</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_2_7_1_1_1"><b>suppose </b></a><a NAME="E1:50_2_1_2_7_1_1_1"/><i><font color="Green" title="E63">A60</font></i>: 
<font color="Maroon" title="c46">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="c28">b1m</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"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c42">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font></span><br/><div class="add"><a NAME="E2:50_2_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E64">A61</font></i>: 
( <font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">b1m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">I</font> &amp; <font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font></span>)</span> &amp; <font color="Maroon" title="c6">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">b1m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">I</font> )
 <b>by </b><i><a class="txt" href="#E14:50_2_1_2"><i><font color="Green" title="E52">A33</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c42">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> = 
<span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c36">B</font>,<font color="Maroon" title="c28">b1m</font>,<font color="Maroon" title="c30">b2m</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="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font></span>)</span>
<b>by </b><i><a class="txt" href="#E34:50_2_1_2"><i><font color="Green" title="E59">A56</font></i></a>, <a class="txt" href="#E14:50_2_1_2"><i><font color="Green" title="E52">A33</font></i></a>, <a class="ref" href="matrixj2.html#T19" target="_self" title="MATRIXJ2:th.19">Th19</a>, <a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="txt" href="#E3:50_2_1_2_7"><i><font color="Green" title="E62">A59</font></i></a>, <a class="txt" href="#E1:50_2_1_2_7_1_1_1"><i><font color="Green" title="E63">A60</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c33">FL</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">b1m</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font></span>)</span>
<b>by </b><i><a class="txt" href="#E28:50_2_1_2"><i><font color="Green" title="E58">A53</font></i></a>, <a class="ref" href="matrlin2.html#T34" title="MATRLIN2:th.34">MATRLIN2:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font>
<b>by </b><i><a class="txt" href="#E2:50_2_1_2_7_1_1_1"><i><font color="Green" title="E64">A61</font></i></a>, <a class="txt" href="#E1:50_2_1_2_7"><i><font color="Green" title="E61">A58</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</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="50_2_1_2_7_1_1_2"><b>suppose </b></a><a NAME="E1:50_2_1_2_7_1_1_2"/>
 ex <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">j</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="c29">b1'm</font> &amp; <font color="Maroon" title="c46">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">b1m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</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="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c42">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c47">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:50_2_1_2_7_1_1_2"/><i><font color="Green" title="E63">A62</font></i>: 
( <font color="Maroon" title="c47">j</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="c29">b1'm</font> &amp; <font color="Maroon" title="c46">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">b1m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c47">j</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_2_7_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_2_7_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_2_1_2_7_1_1_2_1_1"/>
( <font color="Maroon" title="c47">j</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="c29">b1'm</font> or <font color="Maroon" title="c47">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">b1'm</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_2_7_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:50_2_1_2_7_1_1_2_1_1_1"/><i><font color="Green" title="E64">A63</font></i>: 
<font color="Maroon" title="c47">j</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="c29">b1'm</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="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c41">JB</font>,<font color="Maroon" title="c29">b1'm</font>,<font color="Maroon" title="c31">b2'm</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="c29">b1'm</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">j</font></span>)</span></span><br/><div class="add"><a NAME="E2:50_2_1_2_7_1_1_2_1_1_1"/><i><font color="Green" title="E65">A64</font></i>: 
( <font color="Maroon" title="c31">b2'm</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">b2'm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">j</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">b2'm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">j</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">I</font> )
 <b>by </b><i><a class="txt" href="#E10:50_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="#E14:50_2_1_2"><i><font color="Green" title="E52">A33</font></i></a>, <a class="txt" href="#E1:50_2_1_2_7"><i><font color="Green" title="E61">A58</font></i></a>, <a class="txt" href="#E3:50_2_1_2_7_1_1_2"><i><font color="Green" title="E63">A62</font></i></a>, <a class="txt" href="#E15:50_2_1_2"><i><font color="Green" title="E53">A34</font></i></a>, <a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> = 
<font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:50_2_1_2_7"><i><font color="Green" title="E62">A59</font></i></a>, <a class="txt" href="#E1:50_2_1_2_7_1_1_2_1_1_1"><i><font color="Green" title="E64">A63</font></i></a>, <a class="txt" href="#E11:50_2"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="#E4:50_2"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="#E3:50_2_1_2_7_1_1_2"><i><font color="Green" title="E63">A62</font></i></a>, <a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">b2'm</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">j</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:50_2_1_2_7_1_1_2_1_1_1"><i><font color="Green" title="E65">A64</font></i></a>, <a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c41">JB</font>,<font color="Maroon" title="c29">b1'm</font>,<font color="Maroon" title="c31">b2'm</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="c29">b1'm</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">j</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:50_2_1_2_7_1_1_2_1_1_1"><i><font color="Green" title="E64">A63</font></i></a>, <a class="txt" href="#E3:50_2_1_2_7_1_1_2"><i><font color="Green" title="E63">A62</font></i></a>, <a class="ref" href="matrixj2.html#T23" target="_self" title="MATRIXJ2:th.23">Th23</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="50_2_1_2_7_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:50_2_1_2_7_1_1_2_1_1_2"/><i><font color="Green" title="E64">A65</font></i>: 
<font color="Maroon" title="c47">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">b1'm</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="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c41">JB</font>,<font color="Maroon" title="c29">b1'm</font>,<font color="Maroon" title="c31">b2'm</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="c29">b1'm</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">j</font></span>)</span></span><br/><div class="add"><a NAME="E2:50_2_1_2_7_1_1_2_1_1_2"/>
<font color="Maroon" title="c47">j</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> <font color="Maroon" title="c29">b1'm</font>
 <b>by </b><i><a class="txt" href="#E3:50_2_1_2_7_1_1_2"><i><font color="Green" title="E63">A62</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:50_2_1_2_7_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c47">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">b1'm</font>
 <b>by </b><i><a class="txt" href="#E1:50_2_1_2_7_1_1_2_1_1_2"><i><font color="Green" title="E64">A65</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:50_2_1_2_7_1_1_2_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c47">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c47">j</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">b1'm</font> )
 <b>by </b><i><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="E5:50_2_1_2_7_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c47">j</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">b1'm</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:50_2_1_2_7_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c31">b2'm</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">b2'm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">j</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">b2'm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">j</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">I</font> &amp; <font color="Maroon" title="c31">b2'm</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">b2'm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c7">b2</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">b1m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">j</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="c31">b2'm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">b1m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">j</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">b2</font> )
 <b>by </b><i><a class="txt" href="#E10:50_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="#E14:50_2_1_2"><i><font color="Green" title="E52">A33</font></i></a>, <a class="txt" href="#E1:50_2_1_2_7"><i><font color="Green" title="E61">A58</font></i></a>, <a class="txt" href="#E3:50_2_1_2_7_1_1_2"><i><font color="Green" title="E63">A62</font></i></a>, <a class="txt" href="#E15:50_2_1_2"><i><font color="Green" title="E53">A34</font></i></a>, <a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E7:50_2_1_2_7_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">b2'm</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font></span>)</span> &amp; <font color="Maroon" title="c31">b2'm</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c47">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">I</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:50_2_1_2_7_1_1_2"><i><font color="Green" title="E63">A62</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a>, <a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</a></i>;<br/><b>then </b><i><font color="Green" title="E65">A66</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font></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="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c46">I</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">b2'm</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">j</font></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="c31">b2'm</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_4.html#T21" title="VECTSP_4:th.21">VECTSP_4:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c41">JB</font>,<font color="Maroon" title="c29">b1'm</font>,<font color="Maroon" title="c31">b2'm</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="c29">b1'm</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">j</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:50_2_1_2_7_1_1_2"><i><font color="Green" title="E63">A62</font></i></a>, <a class="txt" href="#E1:50_2_1_2_7_1_1_2_1_1_2"><i><font color="Green" title="E64">A65</font></i></a>, <a class="ref" href="matrixj2.html#T23" target="_self" title="MATRIXJ2:th.23">Th23</a></i>
;<br/><a NAME="E9:50_2_1_2_7_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font></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="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c46">I</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_2_7_1_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_2_1_2_7_1_1_2_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font></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="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c46">I</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:50_2_1_2_7_1_1_2_1_1_2_2"/><b>then </b>
( <font color="Maroon" title="c10">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">I</font></span>)</span> &amp; <font color="Maroon" title="c46">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c46">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> <font color="Maroon" title="c6">b1</font> )
 
<b>by </b><i><a class="txt" href="#E4:50_2"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="#E1:50_2_1_2_7"><i><font color="Green" title="E61">A58</font></i></a>, <a class="txt" href="#E3:50_2_1_2_7"><i><font color="Green" title="E62">A59</font></i></a>, <a class="txt" href="#E3:50_2_1_2_7_1_1_2"><i><font color="Green" title="E63">A62</font></i></a>, <a class="txt" href="#E1:50_2_1_2_7_1_1_2_1_1_2"><i><font color="Green" title="E64">A65</font></i></a>, <a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:50_2_1_2_7_1_1_2_1_1_2_2"/><b>then </b>
( <font color="Maroon" title="c46">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">FF</font> &amp; <font color="Maroon" title="c46">I</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font> )
 
<b>by </b><i><a class="txt" href="#E1:50_2_1_2_7"><i><font color="Green" title="E61">A58</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E4:50_2_1_2_7_1_1_2_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">b1m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c47">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">b1m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="#E3:50_2_1_2_7_1_1_2"><i><font color="Green" title="E63">A62</font></i></a>, <a class="txt" href="#E6:50_2_1_2"><i><font color="Green" title="E50">A31</font></i></a>, <a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a></i>;<br/>
<a NAME="E5:50_2_1_2_7_1_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c47">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E6:50_2_1_2_7_1_1_2_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:50_2_1_2_7_1_1_2"><i><font color="Green" title="E63">A62</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</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>hence </b><a NAME="E10:50_2_1_2_7_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c41">JB</font>,<font color="Maroon" title="c29">b1'm</font>,<font color="Maroon" title="c31">b2'm</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="c29">b1'm</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:50_2_1_2_7_1_1_2_1_1_2"><i><font color="Green" title="E65">A66</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/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:50_2_1_2_7_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c42">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font>
 <b>by </b><i><a class="txt" href="#E34:50_2_1_2"><i><font color="Green" title="E59">A56</font></i></a>, <a class="txt" href="#E14:50_2_1_2"><i><font color="Green" title="E52">A33</font></i></a>, <a class="ref" href="matrixj2.html#T19" target="_self" title="MATRIXJ2:th.19">Th19</a>, <a class="txt" href="#E7:50_2_1_2"><i><font color="Green" title="E51">A32</font></i></a>, <a class="txt" href="#E3:50_2_1_2_7"><i><font color="Green" title="E62">A59</font></i></a>, <a class="txt" href="#E3:50_2_1_2_7_1_1_2"><i><font color="Green" title="E63">A62</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/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:50_2_1_2_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c44">F'</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c42">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">x</font>
 ; <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="E40:50_2_1_2"/><b>then </b>
<font color="Maroon" title="c10">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="c42">BB</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font>
 <b>by </b><i><a class="txt" href="#E11:50_2"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="#E38:50_2_1_2"><i><font color="Green" title="E60">A57</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a>, <a class="ref" href="matrlin.html#T26" title="MATRLIN:th.26">MATRLIN:26</a></i>;<br/><b>hence </b><a NAME="E41:50_2_1_2"/>
 <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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="c40">JJJ</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>by </b><i><a class="txt" href="#E34:50_2_1_2"><i><font color="Green" title="E59">A56</font></i></a>, <a class="ref" href="matrlin2.html#T36" title="MATRLIN2:th.36">MATRLIN2:36</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>
<a NAME="E3:50"/><i><font color="Green" title="E37">A67</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>1</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="#E1:50"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="#E2:50"><i><font color="Green" title="E36">A6</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c2">V1</font>, <font color="Maroon" title="c3">V2</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">b2</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V2</font><br/> for <font color="Olive" title="b3">L</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">b1</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="Olive" title="b2">b2</font> holds <br/>for <font color="Olive" title="b4">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>  st ( for <font color="Olive" title="b5">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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="b1">b1</font> or <font color="Olive" title="b4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span> or ( <font color="Olive" title="b5">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">b1</font> &amp; <font color="Olive" title="b4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b5">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Olive" title="b3">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b4">F</font>,<font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</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="b5">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></span><br/><b>let </b><font color="Maroon" title="c4">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">b2</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V2</font><br/> for <font color="Olive" title="b2">L</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</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="Olive" title="b1">b2</font> holds <br/>for <font color="Olive" title="b3">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>  st ( for <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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="Maroon" title="c4">b1</font> or <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span> or ( <font color="Olive" title="b4">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b4">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Olive" title="b2">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b3">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Olive" title="b1">b2</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="b4">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></span><br/><b>let </b><font color="Maroon" title="c5">b2</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V2</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">L</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</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="c5">b2</font> holds <br/>for <font color="Olive" title="b2">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>  st ( for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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">b1</font> or <font color="Olive" title="b2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span> or ( <font color="Olive" title="b3">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Olive" title="b2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></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="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b3">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Olive" title="b1">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b2">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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></span><br/><b>let </b><font color="Maroon" title="c6">L</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</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="c5">b2</font> implies for <font color="Olive" title="b1">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>  st ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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">b1</font> or <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> or ( <font color="Olive" title="b2">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></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="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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> )</span><br/>









<b>assume </b><a NAME="E4:50"/><i><font color="Green" title="E38">A68</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</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="c5">b2</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">F</font> being  <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</font>  st ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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">b1</font> or <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> or ( <font color="Olive" title="b2">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></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="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) holds <br/> ex <font color="Olive" title="b2">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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></span><br/>

<b>let </b><font color="Maroon" title="c7">F</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c3">V2</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">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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">b1</font> or <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> or ( <font color="Olive" title="b1">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></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="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) ) ) implies  ex <font color="Olive" title="b1">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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> )</span><br/>

<b>assume </b><a NAME="E5:50"/><i><font color="Green" title="E39">A69</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <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">b1</font> or <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> or ( <font color="Olive" title="b1">i</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></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="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )
 ; <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">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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></span><br/>

<b>set </b><font color="Maroon" title="c8">FF</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span> ;<br/>
<a NAME="E6:50"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">b1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span>  or <font color="Maroon" title="c9">x</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">b1</font> )</span><br/>

<b>assume </b><a NAME="E1:50_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</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">b1</font></span><br/>

<a NAME="E2:50_3"/><b>then </b>
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp; <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">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:50_3"/>
<font color="Maroon" title="c9">x</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">b1</font>
 ; <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 reconsider </b><font color="Maroon" title="c9">FF</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <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">b1</font> &amp; <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="c4">b1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E8:50"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">FF</font>]
 
<b>by </b><i><a class="txt" href="#E3:50"><i><font color="Green" title="E37">A67</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:50"/>
 ex <font color="Olive" title="b1">J</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="matrixj2.html#M1" title="MATRIXJ2:mode.1">FinSequence_of_Jordan_block</a> of <font color="Maroon" title="c6">L</font>,<font color="Maroon" title="c1">K</font> st  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</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>
 <b>by </b><i><a class="txt" href="#E4:50"><i><font color="Green" title="E38">A68</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E39">A69</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/>


</div>
