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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font>  st  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b4">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">K</font> st <br/>( ( for <font color="Olive" title="b5">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>  ex <font color="Olive" title="b6">Path</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Olive" title="b1">K</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b7">Fj</font>, <font color="Olive" title="b8">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b7">Fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">j</font> holds <br/><font color="Olive" title="b6">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b8">j</font>,<font color="Olive" title="b7">Fj</font> ) &amp; <font color="Olive" title="b4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Olive" title="b6">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b5">F</font></span>)</span></span>)</span> ) ) &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b1">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b4">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b3">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>( ( for <font color="Olive" title="b4">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>  ex <font color="Olive" title="b5">Path</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b6">Fj</font>, <font color="Olive" title="b7">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b6">Fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">j</font> holds <br/><font color="Olive" title="b5">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b7">j</font>,<font color="Olive" title="b6">Fj</font> ) &amp; <font color="Olive" title="b3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Olive" title="b5">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b4">F</font></span>)</span></span>)</span> ) ) &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b3">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> implies  ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>( ( for <font color="Olive" title="b2">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>  ex <font color="Olive" title="b3">Path</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b4">Fj</font>, <font color="Olive" title="b5">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b4">Fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font> holds <br/><font color="Olive" title="b3">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b5">j</font>,<font color="Olive" title="b4">Fj</font> ) &amp; <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Olive" title="b3">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b2">F</font></span>)</span></span>)</span> ) ) &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">P</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:83"/><span class="lab"><font color="Green" title="E71">A1</font></span>: 
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>( ( for <font color="Olive" title="b2">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>  ex <font color="Olive" title="b3">Path</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b4">Fj</font>, <font color="Olive" title="b5">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b4">Fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font> holds <br/><font color="Olive" title="b3">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b5">j</font>,<font color="Olive" title="b4">Fj</font> ) &amp; <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Olive" title="b3">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b2">F</font></span>)</span></span>)</span> ) ) &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">P</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">AB</font> = <font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">aa</font> = the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">I</font> =  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">mm</font> = the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">KK</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>,  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $2</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span><br/> for <font color="Olive" title="b2">M</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b1">F</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $2</span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b3">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $2 implies <font color="Olive" title="b2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font></span>)</span> ) &amp; ( not <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $2 implies <font color="Olive" title="b2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> ) ) ) holds <br/> ex <font color="Olive" title="b3">Path</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; ( for <font color="Olive" title="b4">Fj</font>, <font color="Olive" title="b5">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $2 &amp; <font color="Olive" title="b4">Fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font> holds <br/><font color="Olive" title="b3">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b5">j</font>,<font color="Olive" title="b4">Fj</font> ) &amp; $1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Olive" title="b3">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Olive" title="b2">M</font></span>)</span> );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies for <font color="Olive" title="b1">FUNC</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">FUNC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">FUNC</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">P</font>,$1] &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <font color="Olive" title="b1">FUNC</font></span>)</span>,<font color="Olive" title="b2">P</font> ) );<br/>
<a NAME="E2:83"/><span class="lab"><font color="Green" title="E72">A2</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c10">Y</font> =  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c11">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_1"/><span class="lab"><font color="Green" title="E73">A3</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c12">X</font> =  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">FUNC</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E71">A1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">k1</font> = <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<span class="kw">assume </span><a NAME="E3:83_1"/><span class="lab"><font color="Green" title="E74">A4</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">FUNC</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">FUNC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">FUNC</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">P</font>,<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <font color="Olive" title="b1">FUNC</font></span>)</span>,<font color="Olive" title="b2">P</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c15">Xx</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c16">FUNC1</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">FUNC1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> implies  ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c16">FUNC1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <font color="Maroon" title="c16">FUNC1</font></span>)</span>,<font color="Olive" title="b1">P</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:83_1"/><span class="lab"><font color="Green" title="E75">A5</font></span>: 
<font color="Maroon" title="c16">FUNC1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c16">FUNC1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <font color="Maroon" title="c16">FUNC1</font></span>)</span>,<font color="Olive" title="b1">P</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c17">P1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c16">FUNC1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E6:83_1"/><span class="lab"><font color="Green" title="E76">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">P1</font>,<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="txt" href="#E4:83_1"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="txt" href="matrix11.html#E74"><span class="lab"><font color="Green" title="E69">Lm10</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">FUNC19</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_1"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="ref" href="finseq_1.html#T11" title="FINSEQ_1:th.11">FINSEQ_1:11</a></span>;<br/>
<a NAME="E8:83_1"/><span class="lab"><font color="Green" title="E77">A7</font></span>: 
<font color="Maroon" title="c16">FUNC1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_1"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="ref" href="finseq_1.html#T11" title="FINSEQ_1:th.11">FINSEQ_1:11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">P19</font> = <font color="Maroon" title="c17">P1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c18">FUNC19</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> ;<br/>
<a NAME="E10:83_1"/><span class="lab"><font color="Green" title="E78">A8</font></span>: 
<font color="Maroon" title="c11">k</font> <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>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">P</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c13">FUNC</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E12:83_1"/><span class="lab"><font color="Green" title="E79">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c11">k</font>]
 <span class="kw">and </span><br/><a NAME="E13:83_1"/><span class="lab"><font color="Green" title="E80">A10</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <font color="Maroon" title="c13">FUNC</font></span>)</span>,<font color="Maroon" title="c20">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="txt" href="#E3:83_1"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E14:83_1"/><span class="lab"><font color="Green" title="E81">A11</font></span>: 
not <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T9" title="FINSEQ_3:th.9">FINSEQ_3:9</a></span>;<br/>
<a NAME="E15:83_1"/><span class="lab"><font color="Green" title="E82">A12</font></span>: 
for <font color="Olive" title="b1">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span><br/> for <font color="Olive" title="b2">SF</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c18">FUNC19</font>  st <font color="Olive" title="b2">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">h</font> where <font color="Olive" title="b3">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b3">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H</font> </span>}</span>  holds <br/>the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b2">SF</font>,<font color="Maroon" title="c19">P19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c21">YY</font> =  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E71">A1</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c22">H</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">SF</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c18">FUNC19</font>  st <font color="Olive" title="b1">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">h</font> where <font color="Olive" title="b2">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b2">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font> </span>}</span>  holds <br/>the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b1">SF</font>,<font color="Maroon" title="c19">P19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">H</font></span><br/><span class="kw">let </span><font color="Maroon" title="c23">SF</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c18">FUNC19</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b1">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font> </span>}</span>  implies the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c23">SF</font>,<font color="Maroon" title="c19">P19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">H</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:83_1_1"/><span class="lab"><font color="Green" title="E83">A13</font></span>: 
<font color="Maroon" title="c23">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b1">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c23">SF</font>,<font color="Maroon" title="c19">P19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">H</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">h</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font> &amp; <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2;<br/>
<a NAME="E3:83_1_1"/><span class="lab"><font color="Green" title="E84">A14</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">YY</font> holds <br/> ex <font color="Olive" title="b2">f9</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">f9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">SF</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">f9</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">YY</font> implies  ex <font color="Olive" title="b1">f9</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">f9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">SF</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c24">y</font>,<font color="Olive" title="b1">f9</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:83_1_1_1"/>
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">YY</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f9</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">f9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">SF</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c24">y</font>,<font color="Olive" title="b1">f9</font>] )</span><br/>

<a NAME="E2:83_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c24">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">q</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:83_1_1_1"/><span class="lab"><font color="Green" title="E85">A15</font></span>: 
<font color="Maroon" title="c25">q</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font>
 <span class="kw">and </span><br/><a NAME="E5:83_1_1_1"/><span class="lab"><font color="Green" title="E86">A16</font></span>: 
<font color="Maroon" title="c25">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:83_1"><span class="lab"><font color="Green" title="E81">A11</font></span></a>, <a class="ref" href="stirl2_1.html#T67" title="STIRL2_1:th.67">STIRL2_1:67</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c25">q</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">SF</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c24">y</font>,<font color="Maroon" title="c25">q</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E6:83_1_1_1"/>
<font color="Maroon" title="c25">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">SF</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_1_1"><span class="lab"><font color="Green" title="E83">A13</font></span></a>, <a class="txt" href="#E4:83_1_1_1"><span class="lab"><font color="Green" title="E85">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c24">y</font>,<font color="Maroon" title="c25">q</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c26">h</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font> &amp; <font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">y</font> implies <font color="Maroon" title="c26">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:83_1_1_1"/><span class="lab"><font color="Green" title="E87">A17</font></span>: 
<font color="Maroon" title="c26">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font>
 <span class="kw">and </span><br/><a NAME="E8:83_1_1_1"/><span class="lab"><font color="Green" title="E88">A18</font></span>: 
<font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c27">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:83_1_1_1_1"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font></span><br/><a NAME="E2:83_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A19</font></span>: 
( <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font> or <font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E3:83_1_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c22">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E4:83_1_1_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c25">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font> &amp; <font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font> ) or <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_1_1_1"><span class="lab"><font color="Green" title="E85">A15</font></span></a>, <a class="txt" href="#E7:83_1_1_1"><span class="lab"><font color="Green" title="E87">A17</font></span></a>, <a class="txt" href="#E2:83_1_1_1_1"><span class="lab"><font color="Green" title="E89">A19</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:83_1_1_1_1"/>
<font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1_1"><span class="lab"><font color="Green" title="E86">A16</font></span></a>, <a class="txt" href="#E8:83_1_1_1"><span class="lab"><font color="Green" title="E88">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:83_1_1_1"/>
<font color="Maroon" title="c26">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c24">QQ</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c21">YY</font>,<font color="Maroon" title="c23">SF</font><span class="kw"> such that </span><br/><a NAME="E5:83_1_1"/><span class="lab"><font color="Green" title="E85">A20</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">YY</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c24">QQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E3:83_1_1"><span class="lab"><font color="Green" title="E84">A14</font></span></a>);<br/></span>
<a NAME="E6:83_1_1"/><span class="lab"><font color="Green" title="E86">A21</font></span>: 
<font color="Maroon" title="c23">SF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">QQ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">SF</font> or <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">QQ</font> )</span><br/>

<a NAME="E1:83_1_1_2"/>
<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E2:83_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A22</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:83_1_1_2"/><span class="lab"><font color="Green" title="E88">A23</font></span>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">SF</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">QQ</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:83_1_1_2"/><span class="lab"><font color="Green" title="E89">A24</font></span>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">h</font>
 <span class="kw">and </span><br/><a NAME="E6:83_1_1_2"/><span class="lab"><font color="Green" title="E90">A25</font></span>: 
<font color="Maroon" title="c26">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_1_1"><span class="lab"><font color="Green" title="E83">A13</font></span></a></span>;<br/>
<a NAME="E7:83_1_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c26">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E8:83_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A26</font></span>: 
<font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c26">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_1_1_2"><span class="lab"><font color="Green" title="E87">A22</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E9:83_1_1_2"/><span class="lab"><font color="Green" title="E92">A27</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c26">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E10:83_1_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">QQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">YY</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1_1_2"><span class="lab"><font color="Green" title="E88">A23</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E11:83_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">QQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">QQ</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_1_1_2"><span class="lab"><font color="Green" title="E91">A26</font></span></a>, <a class="txt" href="#E9:83_1_1_2"><span class="lab"><font color="Green" title="E92">A27</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:83_1_1_2"/>
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">QQ</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1"><span class="lab"><font color="Green" title="E85">A20</font></span></a>, <a class="txt" href="#E5:83_1_1_2"><span class="lab"><font color="Green" title="E89">A24</font></span></a>, <a class="txt" href="#E6:83_1_1_2"><span class="lab"><font color="Green" title="E90">A25</font></span></a>, <a class="txt" href="#E8:83_1_1_2"><span class="lab"><font color="Green" title="E91">A26</font></span></a>, <a class="txt" href="#E9:83_1_1_2"><span class="lab"><font color="Green" title="E92">A27</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:83_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">QQ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">SF</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E8:83_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A28</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">QQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">SF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_1_1"><span class="lab"><font color="Green" title="E86">A21</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E9:83_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:83_1_1"/><span class="lab"><font color="Green" title="E88">A29</font></span>: 
<font color="Maroon" title="c25">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font>
 <span class="kw">and </span><br/><a NAME="E12:83_1_1"/>
<font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:83_1"><span class="lab"><font color="Green" title="E81">A11</font></span></a>, <a class="ref" href="stirl2_1.html#T67" title="STIRL2_1:th.67">STIRL2_1:67</a></span>;<br/>
<a NAME="E13:83_1_1"/><span class="lab"><font color="Green" title="E89">A30</font></span>: 
<font color="Maroon" title="c23">SF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">FUNC19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/>
<a NAME="E14:83_1_1"/>
<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="txt" href="#E10:83_1"><span class="lab"><font color="Green" title="E78">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">Mh</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E16:83_1_1"/><span class="lab"><font color="Green" title="E90">A31</font></span>: 
<font color="Maroon" title="c26">Mh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c22">H</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:83_1_1"/><span class="lab"><font color="Green" title="E91">A32</font></span>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font> implies <font color="Maroon" title="c26">Mh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> ) &amp; ( not <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font> implies <font color="Maroon" title="c26">Mh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="ref" href="matrix11.html#T59" target="_self" title="MATRIX11:th.59">Th59</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">Path</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E19:83_1_1"/><span class="lab"><font color="Green" title="E92">A33</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">and </span><br/><a NAME="E20:83_1_1"/><span class="lab"><font color="Green" title="E93">A34</font></span>: 
for <font color="Olive" title="b1">Hj</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Olive" title="b1">Hj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c27">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">Hj</font>
 <span class="kw">and </span><br/><a NAME="E21:83_1_1"/><span class="lab"><font color="Green" title="E94">A35</font></span>: 
<font color="Maroon" title="c20">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c26">Mh</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:83_1"><span class="lab"><font color="Green" title="E79">A9</font></span></a>, <a class="txt" href="#E16:83_1_1"><span class="lab"><font color="Green" title="E90">A31</font></span></a>, <a class="txt" href="#E17:83_1_1"><span class="lab"><font color="Green" title="E91">A32</font></span></a></span>;<br/>
<a NAME="E22:83_1_1"/><span class="lab"><font color="Green" title="E95">A36</font></span>: 
<font color="Maroon" title="c26">Mh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</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="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:83_1"><span class="lab"><font color="Green" title="E81">A11</font></span></a>, <a class="txt" href="#E17:83_1_1"><span class="lab"><font color="Green" title="E91">A32</font></span></a></span>;<br/>
<a NAME="E23:83_1_1"/>
<font color="Maroon" title="c25">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">SF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_1_1"><span class="lab"><font color="Green" title="E83">A13</font></span></a>, <a class="txt" href="#E11:83_1_1"><span class="lab"><font color="Green" title="E88">A29</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">QQ</font> = <font color="Maroon" title="c24">QQ</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c21">YY</font>,<font color="Maroon" title="c18">FUNC19</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_1_1"><span class="lab"><font color="Green" title="E87">A28</font></span></a>, <a class="txt" href="#E13:83_1_1"><span class="lab"><font color="Green" title="E89">A30</font></span></a>, <a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<a NAME="E25:83_1_1"/><span class="lab"><font color="Green" title="E96">A37</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">P19</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">QQ</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E26:83_1_1"/><span class="lab"><font color="Green" title="E97">A38</font></span>: 
<font color="Maroon" title="c28">QQ</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">QQ</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">SF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_1_1"><span class="lab"><font color="Green" title="E87">A28</font></span></a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></span>;<br/>
<a NAME="E27:83_1_1"/>
1 <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>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E28:83_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A39</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1"><span class="lab"><font color="Green" title="E74">A4</font></span></a></span>;<br/>
<a NAME="E29:83_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A40</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</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="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>;<br/>
<a NAME="E30:83_1_1"/>
<font color="Maroon" title="c26">Mh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c26">Mh</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:83_1_1"><span class="lab"><font color="Green" title="E98">A39</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>;<br/>
<a NAME="E31:83_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">Mh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:83_1_1"><span class="lab"><font color="Green" title="E95">A36</font></span></a>, <a class="txt" href="#E29:83_1_1"><span class="lab"><font color="Green" title="E99">A40</font></span></a>, <a class="ref" href="matrix11.html#T30" target="_self" title="MATRIX11:th.30">Th30</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">SUM1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E33:83_1_1"/><span class="lab"><font color="Green" title="E100">A41</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">SUM1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E34:83_1_1"/><span class="lab"><font color="Green" title="E101">A42</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c26">Mh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c29">SUM1</font>
 <span class="kw">and </span><br/><a NAME="E35:83_1_1"/><span class="lab"><font color="Green" title="E102">A43</font></span>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c29">SUM1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Olive" title="b1">j</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:83_1_1"><span class="lab"><font color="Green" title="E98">A39</font></span></a>, <a class="ref" href="matrix11.html#T56" target="_self" title="MATRIX11:th.56">Th56</a></span>;<br/>
<a NAME="E36:83_1_1"/><span class="lab"><font color="Green" title="E103">A44</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c30">PA</font> = the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c31">PS</font> = <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font>;<br/>
<a NAME="E37:83_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:83_1_1"><span class="lab"><font color="Green" title="E100">A41</font></span></a>, <a class="txt" href="matrix11.html#E38"><span class="lab"><font color="Green" title="E36">Lm5</font></span></a></span>;<br/>
<a NAME="E38:83_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A45</font></span>: 
 <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">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c32">PSaa</font> =  <a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font></span>)</span>;<br/>
<a NAME="E39:83_1_1"/><span class="lab"><font color="Green" title="E105">A46</font></span>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c19">P19</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">QQ</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Olive" title="b1">j</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:83_1_1_3"/><span class="lab"><font color="Green" title="E106">A47</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E2:83_1_1_3"/><span class="lab"><font color="Green" title="E107">A48</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">Mh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E3:83_1_1_3"/><span class="lab"><font color="Green" title="E108">A49</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">P19</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">QQ</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:83_1_1_3"/><span class="lab"><font color="Green" title="E109">A50</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c26">Mh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c33">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c33">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c19">P19</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">QQ</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E5:83_1_1_3"/><span class="lab"><font color="Green" title="E110">A51</font></span>: 
<font color="Maroon" title="c33">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c19">P19</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">QQ</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E6:83_1_1_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c33">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1_3"><span class="lab"><font color="Green" title="E110">A51</font></span></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c34">hj</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:83_1_1_3"/><span class="lab"><font color="Green" title="E111">A52</font></span>: 
<font color="Maroon" title="c34">hj</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font>
 <span class="kw">and </span><br/><a NAME="E9:83_1_1_3"/><span class="lab"><font color="Green" title="E112">A53</font></span>: 
<font color="Maroon" title="c34">hj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:83_1"><span class="lab"><font color="Green" title="E81">A11</font></span></a>, <a class="ref" href="stirl2_1.html#T67" title="STIRL2_1:th.67">STIRL2_1:67</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c35">L</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c36">R</font> =  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span>;<br/>
<a NAME="E10:83_1_1_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T11" title="FINSEQ_1:th.11">FINSEQ_1:11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c37">hj9</font> = <font color="Maroon" title="c34">hj</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c38">Mhj</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E13:83_1_1_3"/><span class="lab"><font color="Green" title="E113">A54</font></span>: 
<font color="Maroon" title="c38">Mhj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c37">hj9</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:83_1_1_3"/><span class="lab"><font color="Green" title="E114">A55</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies <font color="Maroon" title="c38">Mhj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">hj9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) &amp; ( not <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies <font color="Maroon" title="c38">Mhj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="ref" href="matrix11.html#T59" target="_self" title="MATRIX11:th.59">Th59</a></span>;<br/>
<a NAME="E15:83_1_1_3"/><span class="lab"><font color="Green" title="E115">A56</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c38">Mhj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E16:83_1_1_3"/><span class="lab"><font color="Green" title="E116">A57</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E117">A58</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_3_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:83_1_1_3_1"/><span class="lab"><font color="Green" title="E118">A59</font></span>: 
<font color="Maroon" title="c11">k</font> <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>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c39">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">i</font> &amp; <font color="Maroon" title="c39">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="c38">Mhj</font> implies <font color="Maroon" title="c38">Mhj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:83_1_1_3_1"/><span class="lab"><font color="Green" title="E119">A60</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">i</font>
 <span class="kw">and </span><br/><a NAME="E3:83_1_1_3_1"/><span class="lab"><font color="Green" title="E120">A61</font></span>: 
<font color="Maroon" title="c39">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="c38">Mhj</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">Mhj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><a NAME="E4:83_1_1_3_1"/><span class="lab"><font color="Green" title="E121">A62</font></span>: 
<font color="Maroon" title="c39">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E5:83_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A63</font></span>: 
<font color="Maroon" title="c39">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:83_1_1_3"><span class="lab"><font color="Green" title="E115">A56</font></span></a>, <a class="txt" href="#E2:83_1_1_3_1"><span class="lab"><font color="Green" title="E119">A60</font></span></a>, <a class="txt" href="#E3:83_1_1_3_1"><span class="lab"><font color="Green" title="E120">A61</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:83_1_1_3_1_1"/>
( <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> or ( <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:83_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E123">A64</font></span>: 
<font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">Mhj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:83_1_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1_3_1"><span class="lab"><font color="Green" title="E118">A59</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:83_1_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A65</font></span>: 
<font color="Maroon" title="c39">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_1_1_3_1"><span class="lab"><font color="Green" title="E119">A60</font></span></a>, <a class="txt" href="#E4:83_1_1_3_1"><span class="lab"><font color="Green" title="E121">A62</font></span></a></span>;<br/><a NAME="E4:83_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E125">A66</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1_3_1"><span class="lab"><font color="Green" title="E122">A63</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>;<br/><a NAME="E5:83_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E126">A67</font></span>: 
<font color="Maroon" title="c39">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_1_1_3_1"><span class="lab"><font color="Green" title="E119">A60</font></span></a>, <a class="txt" href="#E4:83_1_1_3_1"><span class="lab"><font color="Green" title="E121">A62</font></span></a>, <a class="txt" href="#E1:83_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E123">A64</font></span></a></span>;<br/><a NAME="E6:83_1_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A68</font></span>: 
<font color="Maroon" title="c26">Mh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:83_1_1"><span class="lab"><font color="Green" title="E91">A32</font></span></a></span>;<br/><a NAME="E7:83_1_1_3_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c22">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:83_1_1_3"><span class="lab"><font color="Green" title="E115">A56</font></span></a>, <a class="txt" href="#E2:83_1_1_3_1"><span class="lab"><font color="Green" title="E119">A60</font></span></a>, <a class="txt" href="#E3:83_1_1_3_1"><span class="lab"><font color="Green" title="E120">A61</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E8:83_1_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A69</font></span>: 
<font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">hj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_1_1_3"><span class="lab"><font color="Green" title="E111">A52</font></span></a>, <a class="txt" href="#E5:83_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E126">A67</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><a NAME="E9:83_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E129">A70</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c26">Mh</font>,<font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Mh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1_3_1"><span class="lab"><font color="Green" title="E122">A63</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>;<br/><a NAME="E10:83_1_1_3_1_1_1"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c26">Mh</font>,<font color="Maroon" title="c39">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1_3_1"><span class="lab"><font color="Green" title="E122">A63</font></span></a>, <a class="txt" href="#E1:83_1_1_3_1"><span class="lab"><font color="Green" title="E118">A59</font></span></a>, <a class="txt" href="#E1:83_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E123">A64</font></span></a>, <a class="ref" href="matrix11.html#T28" target="_self" title="MATRIX11:th.28">Th28</a></span>;<br/><span class="kw">hence </span><a NAME="E11:83_1_1_3_1_1_1"/>
<font color="Maroon" title="c38">Mhj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:83_1_1_3"><span class="lab"><font color="Green" title="E114">A55</font></span></a>, <a class="txt" href="#E4:83_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E125">A66</font></span></a>, <a class="txt" href="#E9:83_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E129">A70</font></span></a>, <a class="txt" href="#E6:83_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E127">A68</font></span></a>, <a class="txt" href="#E8:83_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E128">A69</font></span></a>, <a class="txt" href="#E3:83_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E124">A65</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:83_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E123">A71</font></span>: 
( <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">Mhj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:83_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E124">A72</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></span>;<br/><a NAME="E3:83_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E125">A73</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1_3"><span class="lab"><font color="Green" title="E110">A51</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>;<br/><a NAME="E4:83_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E126">A74</font></span>: 
<span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c39">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1_3_1"><span class="lab"><font color="Green" title="E122">A63</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>;<br/><a NAME="E5:83_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E127">A75</font></span>: 
<font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E123">A71</font></span></a>, <a class="ref" href="nat_1.html#T9" title="NAT_1:th.9">NAT_1:9</a></span>;<br/><a NAME="E6:83_1_1_3_1_1_2"/><span class="kw">then </span>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c39">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:83_1_1_3"><span class="lab"><font color="Green" title="E116">A57</font></span></a>, <a class="txt" href="#E1:83_1_1_3"><span class="lab"><font color="Green" title="E106">A47</font></span></a>, <a class="txt" href="#E4:83_1_1_3"><span class="lab"><font color="Green" title="E109">A50</font></span></a>, <a class="txt" href="#E5:83_1_1_3_1"><span class="lab"><font color="Green" title="E122">A63</font></span></a>, <a class="ref" href="matrix11.html#T28" target="_self" title="MATRIX11:th.28">Th28</a></span>;<br/><span class="kw">hence </span><a NAME="E7:83_1_1_3_1_1_2"/>
<font color="Maroon" title="c38">Mhj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_1_1_3"><span class="lab"><font color="Green" title="E112">A53</font></span></a>, <a class="txt" href="#E14:83_1_1_3"><span class="lab"><font color="Green" title="E114">A55</font></span></a>, <a class="txt" href="#E5:83_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E127">A75</font></span></a>, <a class="txt" href="#E2:83_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E124">A72</font></span></a>, <a class="txt" href="#E4:83_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E126">A74</font></span></a>, <a class="txt" href="#E3:83_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E125">A73</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:83_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E123">A76</font></span>: 
( <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">Mhj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:83_1_1_3_1_1_3"/><span class="kw">then </span>
not <font color="Maroon" title="c39">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:83_1_1_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A77</font></span>: 
<font color="Maroon" title="c38">Mhj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:83_1_1_3"><span class="lab"><font color="Green" title="E114">A55</font></span></a></span>;<br/><a NAME="E4:83_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E125">A78</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1_3_1"><span class="lab"><font color="Green" title="E122">A63</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>;<br/><a NAME="E5:83_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E126">A79</font></span>: 
not <font color="Maroon" title="c39">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E123">A76</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E6:83_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E127">A80</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c26">Mh</font>,<font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Mh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1_3_1"><span class="lab"><font color="Green" title="E122">A63</font></span></a>, <a class="ref" href="matrix_2.html#T10" title="MATRIX_2:th.10">MATRIX_2:10</a></span>;<br/><a NAME="E7:83_1_1_3_1_1_3"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c26">Mh</font>,<font color="Maroon" title="c39">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1_3_1"><span class="lab"><font color="Green" title="E122">A63</font></span></a>, <a class="txt" href="#E1:83_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E123">A76</font></span></a>, <a class="ref" href="matrix11.html#T28" target="_self" title="MATRIX11:th.28">Th28</a></span>;<br/><span class="kw">hence </span><a NAME="E8:83_1_1_3_1_1_3"/>
<font color="Maroon" title="c38">Mhj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:83_1_1"><span class="lab"><font color="Green" title="E91">A32</font></span></a>, <a class="txt" href="#E4:83_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E125">A78</font></span></a>, <a class="txt" href="#E6:83_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E127">A80</font></span></a>, <a class="txt" href="#E5:83_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E126">A79</font></span></a>, <a class="txt" href="#E3:83_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E124">A77</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E18:83_1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</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="c26">Mh</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="matrix11.html#E34"><span class="lab"><font color="Green" title="E33">Lm4</font></span></a></span>;<br/>
<a NAME="E19:83_1_1_3"/><span class="kw">then </span>
 <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">Mhj</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_1_1_3"><span class="lab"><font color="Green" title="E107">A48</font></span></a>, <a class="txt" href="#E15:83_1_1_3"><span class="lab"><font color="Green" title="E115">A56</font></span></a>, <a class="txt" href="#E17:83_1_1_3"><span class="lab"><font color="Green" title="E117">A58</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c39">Pathj</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E21:83_1_1_3"/><span class="lab"><font color="Green" title="E118">A81</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">Pathj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E22:83_1_1_3"/><span class="lab"><font color="Green" title="E119">A82</font></span>: 
for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">hj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c39">Pathj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">m</font>
 <span class="kw">and </span><br/><a NAME="E23:83_1_1_3"/><span class="lab"><font color="Green" title="E120">A83</font></span>: 
<font color="Maroon" title="c17">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">hj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c39">Pathj</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_1"><span class="lab"><font color="Green" title="E76">A6</font></span></a>, <a class="txt" href="#E13:83_1_1_3"><span class="lab"><font color="Green" title="E113">A54</font></span></a>, <a class="txt" href="#E14:83_1_1_3"><span class="lab"><font color="Green" title="E114">A55</font></span></a></span>;<br/>
<a NAME="E24:83_1_1_3"/><span class="lab"><font color="Green" title="E121">A84</font></span>: 
<font color="Maroon" title="c39">Pathj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c33">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:83_1_1_3"><span class="lab"><font color="Green" title="E112">A53</font></span></a>, <a class="txt" href="#E22:83_1_1_3"><span class="lab"><font color="Green" title="E119">A82</font></span></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E122">A85</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_3_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:83_1_1_3_2"/><span class="lab"><font color="Green" title="E123">A86</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E2:83_1_1_3_2"/><span class="lab"><font color="Green" title="E124">A87</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c22">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1_3"><span class="lab"><font color="Green" title="E110">A51</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c40">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c40">i</font> &amp; <font color="Maroon" title="c40">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="c27">Path</font> implies <font color="Maroon" title="c27">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pathj</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">i</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:83_1_1_3_2"/><span class="lab"><font color="Green" title="E125">A88</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c40">i</font>
 <span class="kw">and </span><br/><a NAME="E4:83_1_1_3_2"/><span class="lab"><font color="Green" title="E126">A89</font></span>: 
<font color="Maroon" title="c40">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="c27">Path</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pathj</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">i</font></span><br/><a NAME="E5:83_1_1_3_2"/><span class="lab"><font color="Green" title="E127">A90</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">Pathj</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Pathj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:83_1_1"><span class="lab"><font color="Green" title="E92">A33</font></span></a>, <a class="txt" href="#E4:83_1_1_3_2"><span class="lab"><font color="Green" title="E126">A89</font></span></a>, <a class="ref" href="finseq_3.html#T121" title="FINSEQ_3:th.121">FINSEQ_3:121</a></span>;<br/><a NAME="E6:83_1_1_3_2"/><span class="lab"><font color="Green" title="E128">A91</font></span>: 
<font color="Maroon" title="c40">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E7:83_1_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A92</font></span>: 
<font color="Maroon" title="c40">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:83_1_1"><span class="lab"><font color="Green" title="E92">A33</font></span></a>, <a class="txt" href="#E3:83_1_1_3_2"><span class="lab"><font color="Green" title="E125">A88</font></span></a>, <a class="txt" href="#E4:83_1_1_3_2"><span class="lab"><font color="Green" title="E126">A89</font></span></a></span>;<br/><a NAME="E8:83_1_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c22">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_1_1_3_2"><span class="lab"><font color="Green" title="E124">A87</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E9:83_1_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1_3_2"><span class="lab"><font color="Green" title="E123">A86</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c41">Hi</font> = <font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">i</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E11:83_1_1_3_2"/>
<font color="Maroon" title="c40">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83_1"><span class="lab"><font color="Green" title="E78">A8</font></span></a>, <a class="txt" href="#E19:83_1_1"><span class="lab"><font color="Green" title="E92">A33</font></span></a>, <a class="txt" href="#E4:83_1_1_3_2"><span class="lab"><font color="Green" title="E126">A89</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:83_1_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A93</font></span>: 
<font color="Maroon" title="c40">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1_1_3_2"><span class="lab"><font color="Green" title="E125">A88</font></span></a>, <a class="txt" href="#E6:83_1_1_3_2"><span class="lab"><font color="Green" title="E128">A91</font></span></a></span>;<br/><a NAME="E13:83_1_1_3_2"/>
<font color="Maroon" title="c40">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:83_1_1"><span class="lab"><font color="Green" title="E92">A33</font></span></a>, <a class="txt" href="#E3:83_1_1_3_2"><span class="lab"><font color="Green" title="E125">A88</font></span></a>, <a class="txt" href="#E4:83_1_1_3_2"><span class="lab"><font color="Green" title="E126">A89</font></span></a>, <a class="txt" href="#E6:83_1_1_3_2"><span class="lab"><font color="Green" title="E128">A91</font></span></a></span>;<br/><a NAME="E14:83_1_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A94</font></span>: 
<font color="Maroon" title="c27">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c40">i</font>,<font color="Maroon" title="c41">Hi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:83_1_1"><span class="lab"><font color="Green" title="E93">A34</font></span></a></span>;<br/><a NAME="E15:83_1_1_3_2"/>
<font color="Maroon" title="c22">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">hj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_1_1_3"><span class="lab"><font color="Green" title="E111">A52</font></span></a>, <a class="txt" href="#E7:83_1_1_3_2"><span class="lab"><font color="Green" title="E129">A92</font></span></a>, <a class="txt" href="#E2:83_1_1_3_2"><span class="lab"><font color="Green" title="E124">A87</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><span class="kw">hence </span><a NAME="E16:83_1_1_3_2"/>
<font color="Maroon" title="c27">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pathj</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:83_1_1_3"><span class="lab"><font color="Green" title="E119">A82</font></span></a>, <a class="txt" href="#E12:83_1_1_3_2"><span class="lab"><font color="Green" title="E130">A93</font></span></a>, <a class="txt" href="#E14:83_1_1_3_2"><span class="lab"><font color="Green" title="E131">A94</font></span></a>, <a class="txt" href="#E5:83_1_1_3_2"><span class="lab"><font color="Green" title="E127">A90</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E26:83_1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">Pathj</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83_1"><span class="lab"><font color="Green" title="E78">A8</font></span></a>, <a class="txt" href="#E21:83_1_1_3"><span class="lab"><font color="Green" title="E118">A81</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/>
<a NAME="E27:83_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c39">Pathj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">Path</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c39">Pathj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:83_1_1"><span class="lab"><font color="Green" title="E92">A33</font></span></a>, <a class="txt" href="#E21:83_1_1_3"><span class="lab"><font color="Green" title="E118">A81</font></span></a>, <a class="txt" href="#E25:83_1_1_3"><span class="lab"><font color="Green" title="E122">A85</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a>, <a class="ref" href="finseq_3.html#T61" title="FINSEQ_3:th.61">FINSEQ_3:61</a></span>;<br/>
<a NAME="E28:83_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A95</font></span>: 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c39">Pathj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c33">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:83_1_1_3"><span class="lab"><font color="Green" title="E121">A84</font></span></a>, <a class="ref" href="finsop_1.html#T5" title="FINSOP_1:th.5">FINSOP_1:5</a></span>;<br/>
<a NAME="E29:83_1_1_3"/>
<font color="Maroon" title="c28">QQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">hj</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1"><span class="lab"><font color="Green" title="E85">A20</font></span></a>, <a class="txt" href="#E5:83_1_1_3"><span class="lab"><font color="Green" title="E110">A51</font></span></a>, <a class="txt" href="#E8:83_1_1_3"><span class="lab"><font color="Green" title="E111">A52</font></span></a>, <a class="txt" href="#E9:83_1_1_3"><span class="lab"><font color="Green" title="E112">A53</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:83_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">P19</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">QQ</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c33">j</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1_3"><span class="lab"><font color="Green" title="E110">A51</font></span></a>, <a class="txt" href="#E23:83_1_1_3"><span class="lab"><font color="Green" title="E120">A83</font></span></a>, <a class="txt" href="#E3:83_1_1_3"><span class="lab"><font color="Green" title="E108">A49</font></span></a>, <a class="txt" href="#E28:83_1_1_3"><span class="lab"><font color="Green" title="E123">A95</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c33">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c33">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="c19">P19</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">QQ</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:83_1_1_4"/><span class="lab"><font color="Green" title="E106">A96</font></span>: 
<font color="Maroon" title="c33">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="c19">P19</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">QQ</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c34">j</font> = <font color="Maroon" title="c33">y</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1_4"><span class="lab"><font color="Green" title="E106">A96</font></span></a></span>;<br/><a NAME="E3:83_1_1_4"/>
<font color="Maroon" title="c29">SUM1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c34">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c34">j</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:83_1_1"><span class="lab"><font color="Green" title="E102">A43</font></span></a>, <a class="txt" href="#E38:83_1_1"><span class="lab"><font color="Green" title="E104">A45</font></span></a>, <a class="txt" href="#E1:83_1_1_4"><span class="lab"><font color="Green" title="E106">A96</font></span></a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</font> = 
<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c34">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p3">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c34">j</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1_4"><span class="lab"><font color="Green" title="E106">A96</font></span></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c34">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <font color="Maroon" title="c26">Mh</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c34">j</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">P19</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">QQ</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:83_1_1"><span class="lab"><font color="Green" title="E105">A46</font></span></a>, <a class="txt" href="#E38:83_1_1"><span class="lab"><font color="Green" title="E104">A45</font></span></a>, <a class="txt" href="#E1:83_1_1_4"><span class="lab"><font color="Green" title="E106">A96</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E41:83_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">P19</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">QQ</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:83_1_1"><span class="lab"><font color="Green" title="E96">A37</font></span></a>, <a class="txt" href="#E38:83_1_1"><span class="lab"><font color="Green" title="E104">A45</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/>
<a NAME="E42:83_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A97</font></span>: 
<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font></span>)</span>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">P19</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">QQ</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setwop_2.html#T23" title="SETWOP_2:th.23">SETWOP_2:23</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c33">x1</font>, <font color="Maroon" title="c34">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c33">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c34">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c28">QQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">QQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x2</font> implies <font color="Maroon" title="c33">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">x2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:83_1_1_5"/><span class="lab"><font color="Green" title="E107">A98</font></span>: 
<font color="Maroon" title="c33">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E2:83_1_1_5"/><span class="lab"><font color="Green" title="E108">A99</font></span>: 
<font color="Maroon" title="c34">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E3:83_1_1_5"/><span class="lab"><font color="Green" title="E109">A100</font></span>: 
<font color="Maroon" title="c28">QQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">QQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c33">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">x2</font></span><br/><a NAME="E4:83_1_1_5"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">x2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:83_1_1_5"><span class="lab"><font color="Green" title="E108">A99</font></span></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></span>;<br/><a NAME="E5:83_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A101</font></span>: 
 ex <font color="Olive" title="b1">h2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">h2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font> &amp; <font color="Olive" title="b1">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">x2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:83_1"><span class="lab"><font color="Green" title="E81">A11</font></span></a>, <a class="ref" href="stirl2_1.html#T67" title="STIRL2_1:th.67">STIRL2_1:67</a></span>;<br/><a NAME="E6:83_1_1_5"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c33">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83_1_1_5"><span class="lab"><font color="Green" title="E107">A98</font></span></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c35">h1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:83_1_1_5"/><span class="lab"><font color="Green" title="E111">A102</font></span>: 
<font color="Maroon" title="c35">h1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">H</font>
 <span class="kw">and </span><br/><a NAME="E9:83_1_1_5"/><span class="lab"><font color="Green" title="E112">A103</font></span>: 
<font color="Maroon" title="c35">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:83_1"><span class="lab"><font color="Green" title="E81">A11</font></span></a>, <a class="ref" href="stirl2_1.html#T67" title="STIRL2_1:th.67">STIRL2_1:67</a></span>;<br/><a NAME="E10:83_1_1_5"/>
<font color="Maroon" title="c28">QQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1"><span class="lab"><font color="Green" title="E85">A20</font></span></a>, <a class="txt" href="#E1:83_1_1_5"><span class="lab"><font color="Green" title="E107">A98</font></span></a>, <a class="txt" href="#E8:83_1_1_5"><span class="lab"><font color="Green" title="E111">A102</font></span></a>, <a class="txt" href="#E9:83_1_1_5"><span class="lab"><font color="Green" title="E112">A103</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:83_1_1_5"/>
<font color="Maroon" title="c33">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:83_1_1"><span class="lab"><font color="Green" title="E85">A20</font></span></a>, <a class="txt" href="#E2:83_1_1_5"><span class="lab"><font color="Green" title="E108">A99</font></span></a>, <a class="txt" href="#E3:83_1_1_5"><span class="lab"><font color="Green" title="E109">A100</font></span></a>, <a class="txt" href="#E9:83_1_1_5"><span class="lab"><font color="Green" title="E112">A103</font></span></a>, <a class="txt" href="#E5:83_1_1_5"><span class="lab"><font color="Green" title="E110">A101</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E44:83_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A104</font></span>: 
<font color="Maroon" title="c28">QQ</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c33">Y9</font> =  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c21">YY</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/>
<a NAME="E46:83_1_1"/><span class="lab"><font color="Green" title="E108">A105</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">QQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">Y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E47:83_1_1"/><span class="lab"><font color="Green" title="E109">A106</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>;<br/>
<a NAME="E48:83_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">P19</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">QQ</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">P19</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">QQ</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:83_1_1"><span class="lab"><font color="Green" title="E96">A37</font></span></a>, <a class="ref" href="relat_1.html#T78" title="RELAT_1:th.78">RELAT_1:78</a></span>;<br/>
<span class="kw">then </span>the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c33">Y9</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">P19</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">QQ</font></span>)</span> = 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K4" title="FINSOP_1:func.4">findom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1" title="SETWOP_2:func.1">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font></span>)</span>,<span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:83_1_1"><span class="lab"><font color="Green" title="E104">A45</font></span></a>, <a class="txt" href="#E36:83_1_1"><span class="lab"><font color="Green" title="E103">A44</font></span></a>, <a class="txt" href="#E47:83_1_1"><span class="lab"><font color="Green" title="E109">A106</font></span></a>, <a class="txt" href="#E42:83_1_1"><span class="lab"><font color="Green" title="E106">A97</font></span></a>, <a class="ref" href="setwop_2.html#T7" title="SETWOP_2:th.7">SETWOP_2:7</a></span>
<br/>.= 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c29">SUM1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T10" title="FVSUM_1:th.10">FVSUM_1:10</a>, <a class="ref" href="setwop_2.html#D2" title="SETWOP_2:def.2">SETWOP_2:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c27">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c29">SUM1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T92" title="FVSUM_1:th.92">FVSUM_1:92</a></span>
<br/>.= 
<font color="Maroon" title="c20">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">H</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:83_1_1"><span class="lab"><font color="Green" title="E94">A35</font></span></a>, <a class="txt" href="#E34:83_1_1"><span class="lab"><font color="Green" title="E101">A42</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E50:83_1_1"/>
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c23">SF</font>,<font color="Maroon" title="c19">P19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:83_1_1"><span class="lab"><font color="Green" title="E107">A104</font></span></a>, <a class="txt" href="#E26:83_1_1"><span class="lab"><font color="Green" title="E97">A38</font></span></a>, <a class="txt" href="#E46:83_1_1"><span class="lab"><font color="Green" title="E108">A105</font></span></a>, <a class="ref" href="setwop_2.html#T8" title="SETWOP_2:th.8">SETWOP_2:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:83_1"/>
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T10" title="FVSUM_1:th.10">FVSUM_1:10</a></span>;<br/>
<a NAME="E17:83_1"/><span class="kw">then </span>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <font color="Maroon" title="c18">FUNC19</font></span>)</span>,<font color="Maroon" title="c19">P19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="txt" href="#E13:83_1"><span class="lab"><font color="Green" title="E80">A10</font></span></a>, <a class="txt" href="#E14:83_1"><span class="lab"><font color="Green" title="E81">A11</font></span></a>, <a class="txt" href="#E15:83_1"><span class="lab"><font color="Green" title="E82">A12</font></span></a>, <a class="ref" href="matrix11.html#T58" target="_self" title="MATRIX11:th.58">Th58</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:83_1"/>
 ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c16">FUNC1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <font color="Maroon" title="c16">FUNC1</font></span>)</span>,<font color="Olive" title="b1">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_1"><span class="lab"><font color="Green" title="E76">A6</font></span></a>, <a class="txt" href="#E8:83_1"><span class="lab"><font color="Green" title="E77">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c10">FUN</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<a NAME="E3:83"/><span class="lab"><font color="Green" title="E73">A107</font></span>: 
<font color="Maroon" title="c1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E4:83"/><span class="lab"><font color="Green" title="E74">A108</font></span>: 
<font color="Maroon">S<sub>2</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="83_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c11">E</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:83_2"/>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">FUNC</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">FUNC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">FUNC</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">P</font>, <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ] &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <font color="Olive" title="b1">FUNC</font></span>)</span>,<font color="Olive" title="b2">P</font> )</span><br/>

<a NAME="E3:83_2"/><span class="lab"><font color="Green" title="E75">A109</font></span>: 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T33" title="GROUP_1:th.33">GROUP_1:33</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">FUNC</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">FUNC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> implies  ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c12">FUNC</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">P</font>, <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ] &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <font color="Maroon" title="c12">FUNC</font></span>)</span>,<font color="Olive" title="b1">P</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:83_2"/><span class="lab"><font color="Green" title="E76">A110</font></span>: 
<font color="Maroon" title="c12">FUNC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c12">FUNC</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">P</font>, <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ] &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <font color="Maroon" title="c12">FUNC</font></span>)</span>,<font color="Olive" title="b1">P</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c13">P</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c12">FUNC</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E6:83_2"/><span class="lab"><font color="Green" title="E77">A111</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">P</font>, <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="txt" href="#E4:83_2"><span class="lab"><font color="Green" title="E76">A110</font></span></a>, <a class="txt" href="matrix11.html#E74"><span class="lab"><font color="Green" title="E69">Lm10</font></span></a></span>;<br/>
<a NAME="E7:83_2"/><span class="lab"><font color="Green" title="E78">A112</font></span>: 
<font color="Maroon" title="c12">FUNC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">E</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83_2"><span class="lab"><font color="Green" title="E76">A110</font></span></a>, <a class="ref" href="funct_5.html#T64" title="FUNCT_5:th.64">FUNCT_5:64</a></span>;<br/>
<a NAME="E8:83_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A113</font></span>: 
<font color="Maroon" title="c11">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">FUNC</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E9:83_2"/>
 <a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <font color="Maroon" title="c12">FUNC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">E</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_2"><span class="lab"><font color="Green" title="E78">A112</font></span></a>, <a class="ref" href="matrix_2.html#D17" title="MATRIX_2:def.17">MATRIX_2:def 17</a></span>;<br/>
<a NAME="E10:83_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A114</font></span>: 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <font color="Maroon" title="c12">FUNC</font></span>)</span>,<font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83_2"><span class="lab"><font color="Green" title="E79">A113</font></span></a>, <a class="ref" href="setwiseo.html#T26" title="SETWISEO:th.26">SETWISEO:26</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">M</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E12:83_2"/><span class="lab"><font color="Green" title="E81">A115</font></span>: 
<font color="Maroon" title="c14">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c11">E</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E13:83_2"/><span class="lab"><font color="Green" title="E82">A116</font></span>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <font color="Maroon" title="c14">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> ) &amp; ( not <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <font color="Maroon" title="c14">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="ref" href="matrix11.html#T59" target="_self" title="MATRIX11:th.59">Th59</a></span>;<br/>
<a NAME="E14:83_2"/><span class="lab"><font color="Green" title="E83">A117</font></span>: 
<font color="Maroon" title="c14">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:83_2"><span class="lab"><font color="Green" title="E81">A115</font></span></a>, <a class="ref" href="funct_4.html#T22" title="FUNCT_4:th.22">FUNCT_4:22</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">Path</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E16:83_2"/><span class="lab"><font color="Green" title="E84">A118</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">and </span><br/><a NAME="E17:83_2"/>
for <font color="Olive" title="b1">Fj</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b1">Fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c15">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">Fj</font>
 <span class="kw">and </span><br/><a NAME="E18:83_2"/><span class="lab"><font color="Green" title="E85">A119</font></span>: 
<font color="Maroon" title="c13">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c15">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c14">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_2"><span class="lab"><font color="Green" title="E77">A111</font></span></a>, <a class="txt" href="#E12:83_2"><span class="lab"><font color="Green" title="E81">A115</font></span></a>, <a class="txt" href="#E13:83_2"><span class="lab"><font color="Green" title="E82">A116</font></span></a></span>;<br/>
<a NAME="E19:83_2"/>
<font color="Maroon" title="c15">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:83_2"><span class="lab"><font color="Green" title="E84">A118</font></span></a></span>;<br/>
<a NAME="E20:83_2"/><span class="kw">then </span>
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c15">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_2"><span class="lab"><font color="Green" title="E75">A109</font></span></a>, <a class="ref" href="finsop_1.html#T11" title="FINSOP_1:th.11">FINSOP_1:11</a></span>;<br/>
<a NAME="E21:83_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:83_2"><span class="lab"><font color="Green" title="E85">A119</font></span></a>, <a class="txt" href="#E14:83_2"><span class="lab"><font color="Green" title="E83">A117</font></span></a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:83_2"/>
 ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c12">FUNC</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">P</font>, <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ] &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <font color="Maroon" title="c12">FUNC</font></span>)</span>,<font color="Olive" title="b1">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_2"><span class="lab"><font color="Green" title="E77">A111</font></span></a>, <a class="txt" href="#E10:83_2"><span class="lab"><font color="Green" title="E80">A114</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:83"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E74">A108</font></span></a>, <a class="txt" href="#E2:83"><span class="lab"><font color="Green" title="E72">A2</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">P</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E7:83"/><span class="lab"><font color="Green" title="E75">A120</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">P</font>,<font color="Maroon" title="c1">n</font>]
 <span class="kw">and </span><br/><a NAME="E8:83"/><span class="lab"><font color="Green" title="E76">A121</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83"><span class="lab"><font color="Green" title="E73">A107</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">P</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>  ex <font color="Olive" title="b2">Path</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b3">Fj</font>, <font color="Olive" title="b4">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">Fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> holds <br/><font color="Olive" title="b2">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">j</font>,<font color="Olive" title="b3">Fj</font> ) &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Olive" title="b2">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b1">F</font></span>)</span></span>)</span> ) ) &amp;  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">P</font> )</span><br/>

<span class="kw">thus </span><a NAME="E9:83"/>
for <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>  ex <font color="Olive" title="b2">Path</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b3">Fj</font>, <font color="Olive" title="b4">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">Fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> holds <br/><font color="Olive" title="b2">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">j</font>,<font color="Olive" title="b3">Fj</font> ) &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Olive" title="b2">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b1">F</font></span>)</span></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">P</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:83_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E2:83_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A122</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Path</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b2">Fj</font>, <font color="Olive" title="b3">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">Fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> holds <br/><font color="Olive" title="b1">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j</font>,<font color="Olive" title="b2">Fj</font> ) &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Olive" title="b1">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span></span>)</span> )</span><br/>

<a NAME="E3:83_3"/><span class="lab"><font color="Green" title="E78">A123</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></span>;<br/>
<a NAME="E4:83_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></span>;<br/>
<a NAME="E5:83_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A124</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_3"><span class="lab"><font color="Green" title="E78">A123</font></span></a>, <a class="ref" href="funct_4.html#T20" title="FUNCT_4:th.20">FUNCT_4:20</a></span>;<br/>
<a NAME="E6:83_3"/><span class="lab"><font color="Green" title="E80">A125</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E7:83_3"/><span class="lab"><font color="Green" title="E81">A126</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c12">F</font></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="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#D4" target="_self" title="MATRIX11:def.4">Def4</a></span>;<br/>
<a NAME="E8:83_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A127</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83_3"><span class="lab"><font color="Green" title="E80">A125</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">M</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E10:83_3"/><span class="lab"><font color="Green" title="E83">A128</font></span>: 
<font color="Maroon" title="c13">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c12">F</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:83_3"/><span class="lab"><font color="Green" title="E84">A129</font></span>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> ) &amp; ( not <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="ref" href="matrix11.html#T59" target="_self" title="MATRIX11:th.59">Th59</a></span>;<br/>
<a NAME="E12:83_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c12">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83_3"><span class="lab"><font color="Green" title="E81">A126</font></span></a>, <a class="txt" href="#E6:83_3"><span class="lab"><font color="Green" title="E80">A125</font></span></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></span>;<br/>
<a NAME="E13:83_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c12">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:83_3"><span class="lab"><font color="Green" title="E83">A128</font></span></a>, <a class="txt" href="#E5:83_3"><span class="lab"><font color="Green" title="E79">A124</font></span></a>, <a class="txt" href="#E8:83_3"><span class="lab"><font color="Green" title="E82">A127</font></span></a>, <a class="txt" href="#E2:83_3"><span class="lab"><font color="Green" title="E77">A122</font></span></a>, <a class="ref" href="funct_4.html#T20" title="FUNCT_4:th.20">FUNCT_4:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:83_3"/>
 ex <font color="Olive" title="b1">Path</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b2">Fj</font>, <font color="Olive" title="b3">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">Fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> holds <br/><font color="Olive" title="b1">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j</font>,<font color="Olive" title="b2">Fj</font> ) &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Olive" title="b1">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:83"><span class="lab"><font color="Green" title="E75">A120</font></span></a>, <a class="txt" href="#E10:83_3"><span class="lab"><font color="Green" title="E83">A128</font></span></a>, <a class="txt" href="#E11:83_3"><span class="lab"><font color="Green" title="E84">A129</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E10:83"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c11">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:83"><span class="lab"><font color="Green" title="E76">A121</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
