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<span class="kw">let </span><font color="Maroon" title="c1">X</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"> for <font color="Olive" title="b1">A</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st not <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/>for <font color="Olive" title="b3">B</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">A</font> holds <br/>for <font color="Olive" title="b4">R</font> being  <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b4">R</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Olive" title="b3">B</font> holds <br/>for <font color="Olive" title="b5">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="Olive" title="b5">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#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b4">R</font>,<font color="Olive" title="b3">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b4">R</font>,<font color="Olive" title="b3">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">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="Olive" title="b2">a</font> holds <br/> <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b4">R</font>,<font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b4">R</font>,<font color="Olive" title="b1">A</font></span>)</span>,<font color="Olive" title="b5">k</font>,<font color="Olive" title="b2">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</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">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/>for <font color="Olive" title="b2">B</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">A</font> holds <br/>for <font color="Olive" title="b3">R</font> being  <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">R</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Olive" title="b2">B</font> holds <br/>for <font color="Olive" title="b4">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="Olive" title="b4">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#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b3">R</font>,<font color="Olive" title="b2">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b3">R</font>,<font color="Olive" title="b2">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">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="Olive" title="b1">a</font> holds <br/> <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b3">R</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b3">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Olive" title="b4">k</font>,<font color="Olive" title="b1">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> implies for <font color="Olive" title="b1">B</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">A</font> holds <br/>for <font color="Olive" title="b2">R</font> being  <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">R</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Olive" title="b1">B</font> holds <br/>for <font color="Olive" title="b3">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="Olive" title="b3">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#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b2">R</font>,<font color="Olive" title="b1">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b2">R</font>,<font color="Olive" title="b1">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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> holds <br/> <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b2">R</font>,<font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b2">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Olive" title="b3">k</font>,<font color="Maroon" title="c3">a</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:19"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</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">B</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">A</font> holds <br/>for <font color="Olive" title="b2">R</font> being  <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">R</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Olive" title="b1">B</font> holds <br/>for <font color="Olive" title="b3">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="Olive" title="b3">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#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b2">R</font>,<font color="Olive" title="b1">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b2">R</font>,<font color="Olive" title="b1">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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> holds <br/> <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b2">R</font>,<font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b2">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Olive" title="b3">k</font>,<font color="Maroon" title="c3">a</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">B</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</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="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">A</font> implies for <font color="Olive" title="b1">R</font> being  <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">R</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c4">B</font> holds <br/>for <font color="Olive" title="b2">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="Olive" title="b2">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#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b1">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b1">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">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> holds <br/> <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b1">R</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b1">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Olive" title="b2">k</font>,<font color="Maroon" title="c3">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:19"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">A</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">R</font> being  <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">R</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c4">B</font> holds <br/>for <font color="Olive" title="b2">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="Olive" title="b2">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#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b1">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b1">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">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> holds <br/> <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b1">R</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Olive" title="b1">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Olive" title="b2">k</font>,<font color="Maroon" title="c3">a</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">R</font> be   <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <font color="Maroon" title="c1">X</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="c5">R</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c4">B</font> implies 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="Olive" title="b1">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#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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> holds <br/> <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c3">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:19"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c5">R</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c4">B</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">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="Olive" title="b1">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#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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> holds <br/> <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c3">a</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">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" title="c6">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#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> implies  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:19"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c6">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#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:19"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">sgb</font> =  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">sga</font> =  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font>;<br/>
<a NAME="E6:19"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
 
<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/>
<a NAME="E7:19"/><span class="kw">then </span>
<font color="Maroon" title="c6">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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E8:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E9:19"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<span class="lab"><font color="Green" title="E24">A9</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="pre_poly.html#T11" title="PRE_POLY:th.11">PRE_POLY:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="polynom2.html#E8"><span class="lab"><font color="Green" title="E6">Lm4</font></span></a>, <a class="ref" href="pre_poly.html#T11" title="PRE_POLY:th.11">PRE_POLY:11</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E25">A10</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:19"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span>
<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/>
<a NAME="E12:19"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="c9">i</font> &amp; <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> implies <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<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="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19_3"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E2:19_3"/><span class="lab"><font color="Green" title="E28">A13</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</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"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<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="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/>

<a NAME="E3:19_3"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon" title="c9">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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_3"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E2:19_3"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E4:19_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 
<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/>
<a NAME="E5:19_3"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:19"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E3:19_3"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_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="E1:19_3_1"/>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c9">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">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="19_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:19_3_1_1"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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"> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<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="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/><div class="add"><span class="kw">thus </span><span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<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="c9">i</font> = 
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_3"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="#E5:19"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E10:19"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E8:19"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E1:19_3_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="finseq_5.html#T76" title="FINSEQ_5:th.76">FINSEQ_5:76</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_3"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E1:19_3_1_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:19_3_1_2"/><span class="lab"><font color="Green" title="E32">A18</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">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"> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<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="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:19_3_1_2_1_1"/>
( <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c6">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="c9">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:19_3_1_2_1_1_1"/>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">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"> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<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="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/><div class="add"><a NAME="E2:19_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A19</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:19_3_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <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="c6">k</font></span>)</span> is   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T19" title="FINSEQ_1:th.19">FINSEQ_1:19</a></span>;<br/><a NAME="E4:19_3_1_2_1_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <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="c6">k</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="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:19"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E8:19"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></span>;<br/><a NAME="E5:19_3_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <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="c6">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_3"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E2:19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E6:19_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A20</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<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="c9">i</font> = 
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_3"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E9:19"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E1:19_3"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="txt" href="#E3:19_3"><span class="lab"><font color="Green" title="E29">A14</font></span></a>, <a class="txt" href="#E2:19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="ref" href="polynom2.html#T11" target="_self" title="POLYNOM2:th.11">Th11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="finseq_5.html#T75" title="FINSEQ_5:th.75">FINSEQ_5:75</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_3"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:19_3_1_2_1_1_2"/><span class="lab"><font color="Green" title="E33">A21</font></span>: 
<font color="Maroon" title="c6">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="c9">i</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"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<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="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/><div class="add"><a NAME="E2:19_3_1_2_1_1_2"/>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_3"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">i9</font> = <font color="Maroon" title="c9">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><a NAME="E4:19_3_1_2_1_1_2"/><span class="lab"><font color="Green" title="E34">A22</font></span>: 
<font color="Maroon" title="c10">i9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E5:19_3_1_2_1_1_2"/>
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_3_1_2"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E1:19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:19_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A23</font></span>: 
<font color="Maroon" title="c6">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="c10">i9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:19_3_1_2_1_1_2"/><span class="lab"><font color="Green" title="E36">A24</font></span>: 
<font color="Maroon" title="c10">i9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_3"><span class="lab"><font color="Green" title="E28">A13</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<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="c9">i</font> = 
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i9</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="#E4:19_3"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E10:19"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E7:19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E36">A24</font></span></a>, <a class="txt" href="#E6:19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="polynom2.html#T12" target="_self" title="POLYNOM2:th.12">Th12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i9</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="#E10:19"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E7:19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E36">A24</font></span></a>, <a class="txt" href="#E6:19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="finseq_5.html#T77" title="FINSEQ_5:th.77">FINSEQ_5:77</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_3"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:19_3_1_2"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<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="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 ; <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="E13:19"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</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> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:19"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:19"/>
 <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:19"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</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>
