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<b>let </b><font color="Maroon" title="c1">m</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Olive" title="b2">x</font>,<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Olive" title="b4">i</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/>( <font color="Olive" title="b3">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; ( for <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b5">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">j</font> &amp; <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Olive" title="b3">y</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>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">K</font> ) )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><br/> for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Olive" title="b3">i</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/>( <font color="Olive" title="b2">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; ( for <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Olive" title="b2">y</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>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Olive" title="b1">i</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/>( <font color="Maroon" title="c4">y</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="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; ( for <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c4">y</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>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) )</span><br/><b>let </b><font color="Maroon" title="c5">i</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> implies ( <font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c4">y</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>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) ) )</span><br/>







<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E31">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c4">y</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>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) )</span><br/>

<a NAME="E2:40"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:40"/><i><font color="Green" title="E33">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E4:40"/><i><font color="Green" title="E34">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:40"/><i><font color="Green" title="E35">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E6:40"/><i><font color="Green" title="E36">B1</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E7:40"/><i><font color="Green" title="E37">B2</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="txt" href="#E5:40"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="#E4:40"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="#E6:40"><i><font color="Green" title="E36">B1</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E8:40"/><i><font color="Green" title="E38">B3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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="funcop_1.html#K3" title="FUNCOP_1:func.3">.:</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:40"><i><font color="Green" title="E37">B2</font></i></a>, <a class="ref" href="funcop_1.html#T84" title="FUNCOP_1:th.84">FUNCOP_1:84</a></i>;<br/>
<i><font color="Green" title="E39">A5</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E8:40"><i><font color="Green" title="E38">B3</font></i></a>, <a class="ref" href="fvsum_1.html#D7" title="FVSUM_1:def.7">FVSUM_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">m</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:40"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font>

;<br/>
<a NAME="E10:40"/><b>then </b><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:40"/><i><font color="Green" title="E41">C1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E12:40"/><i><font color="Green" title="E42">A32</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="txt" href="#E11:40"><i><font color="Green" title="E41">C1</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E10:40"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="matrix14.html#T4" target="_self" title="MATRIX14:th.4">FV73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="#E1:40"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="#E12:40"><i><font color="Green" title="E42">A32</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="#E2:40"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E14:40"/>
<font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E31">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">j</font> being   <a href="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="Olive" title="b1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c4">y</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>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/>

<b>let </b><font color="Maroon" title="c6">j</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> implies <font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> )</span><br/>

<b>assume </b><a NAME="E15:40"/><i><font color="Green" title="E43">B1</font></i>: 
( <font color="Maroon" title="c6">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/>

<a NAME="E16:40"/><i><font color="Green" title="E44">C2</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E15:40"><i><font color="Green" title="E43">B1</font></i></a>, <a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<i><font color="Green" title="E45">C3</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
<b>by </b><i><a class="txt" href="#E16:40"><i><font color="Green" title="E44">C2</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="#E15:40"><i><font color="Green" title="E43">B1</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="#E1:40"><i><font color="Green" title="E31">A1</font></i></a></i>
;<br/>
<a NAME="E18:40"/><i><font color="Green" title="E46">A4b</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E9:40"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="#E15:40"><i><font color="Green" title="E43">B1</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j</font></span>)</span>
<b>by </b><i><a class="txt" href="#E18:40"><i><font color="Green" title="E46">A4b</font></i></a>, <a class="ref" href="matrix14.html#T4" target="_self" title="MATRIX14:th.4">FV73</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="#E17:40"><i><font color="Green" title="E45">C3</font></i></a>, <a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/>
<b>hence </b><a NAME="E20:40"/>
<font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E31">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
