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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b5">k</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="triang_1.html#K2" title="TRIANG_1:func.2">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="triang_1.html#K2" title="TRIANG_1:func.2">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> <font color="Olive" title="b5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> holds <br/>for <font color="Olive" title="b6">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">i</font> &amp; <font color="Olive" title="b6">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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Olive" title="b4">R</font>,<font color="Olive" title="b1">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="b6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Olive" title="b4">R</font>,<font color="Olive" title="b1">A</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b6">i</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">k</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="triang_1.html#K2" title="TRIANG_1:func.2">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="triang_1.html#K2" title="TRIANG_1:func.2">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> <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/>for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Olive" title="b3">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="b5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Olive" title="b3">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">i</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">k</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="triang_1.html#K2" title="TRIANG_1:func.2">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="triang_1.html#K2" title="TRIANG_1:func.2">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> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> holds <br/>for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">k</font> <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Olive" title="b2">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="b4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Olive" title="b2">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font> )</span><br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E15">A1</font></i>: 
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:()"><i><font color="Red">::  thesis: </font></i></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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">k</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="triang_1.html#K2" title="TRIANG_1:func.2">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="triang_1.html#K2" title="TRIANG_1:func.2">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> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> holds <br/>for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">k</font> <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Olive" title="b2">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="b4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Olive" title="b2">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">i</font></span><br/>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k</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="triang_1.html#K2" title="TRIANG_1:func.2">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="triang_1.html#K2" title="TRIANG_1:func.2">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> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> holds <br/>for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k</font> <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Olive" title="b1">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Olive" title="b1">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font> )</span><br/>

<b>assume </b><a NAME="E2:21"/><i><font color="Green" title="E16">A2</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k</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="triang_1.html#K2" title="TRIANG_1:func.2">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="triang_1.html#K2" title="TRIANG_1:func.2">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> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> holds <br/>for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k</font> <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Olive" title="b1">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Olive" title="b1">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span><br/>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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="triang_1.html#K2" title="TRIANG_1:func.2">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="triang_1.html#K2" title="TRIANG_1:func.2">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="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> holds <br/>for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="Olive" title="b2">i</font> )</span><br/>

<b>assume </b><a NAME="E3:21"/><i><font color="Green" title="E17">A3</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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="triang_1.html#K2" title="TRIANG_1:func.2">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="triang_1.html#K2" title="TRIANG_1:func.2">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="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> holds <br/>for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="Olive" title="b2">i</font></span><br/>

<b>let </b><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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">k</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="triang_1.html#K2" title="TRIANG_1:func.2">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="triang_1.html#K2" title="TRIANG_1:func.2">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="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> implies for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">k</font> <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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="Olive" title="b1">i</font> )</span><br/>

<b>assume </b><a NAME="E4:21"/><i><font color="Green" title="E18">A4</font></i>: 
( <font color="Maroon" title="c6">k</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="triang_1.html#K2" title="TRIANG_1:func.2">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="triang_1.html#K2" title="TRIANG_1:func.2">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="c6">k</font> <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:()"><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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">k</font> <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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="Olive" title="b1">i</font></span><br/>

<b>let </b><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> implies <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c7">i</font> )</span><br/>

<b>assume </b><a NAME="E5:21"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c7">i</font></span><br/>

<b>set </b><font color="Maroon" title="c8">sgb</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font>;<br/>
<b>set </b><font color="Maroon" title="c9">sga</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font>;<br/>
<a NAME="E6:21"/><i><font color="Green" title="E20">A6</font></i>: 
<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="c2">A</font>
 
<b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="polynom2.html#E8"><i><font color="Green" title="E6">Lm4</font></i></a></i>;<br/>
<a NAME="E7:21"/>
<font color="Maroon" title="c6">k</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:21"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:21"/><b>then </b><i><font color="Green" title="E21">A7</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:21"/><b>then </b>
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="c6">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">k'</font> = <font color="Maroon" title="c6">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E11:21"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c10">k'</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="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/>
<b>consider </b><font color="Maroon" title="c11">lensga</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E13:21"/><i><font color="Green" title="E23">A9</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">lensga</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">lensgb</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E15:21"/><i><font color="Green" title="E24">A10</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <font color="Maroon" title="c12">lensgb</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">lensga</font> = <font color="Maroon" title="c11">lensga</font>, <font color="Maroon" title="c14">lensgb</font> = <font color="Maroon" title="c12">lensgb</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<i><font color="Green" title="E25">A11</font></i>: <font color="Maroon" title="c14">lensgb</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
<b>by </b><i><a class="txt" href="#E15:21"><i><font color="Green" title="E24">A10</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">B</font>
<b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="triang_1.html#T9" title="TRIANG_1:th.9">TRIANG_1:9</a></i>
<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
<b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>
<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="triang_1.html#K2" title="TRIANG_1:func.2">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
<b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="polynom2.html#E8"><i><font color="Green" title="E6">Lm4</font></i></a>, <a class="ref" href="triang_1.html#T9" title="TRIANG_1:th.9">TRIANG_1:9</a></i>
<br/>.= 
<font color="Maroon" title="c13">lensga</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="#E13:21"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E18:21"/><b>then </b><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c13">lensga</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">lensgb</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E19:21"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></i>;<br/>
<a NAME="E20:21"/><b>then </b><i><font color="Green" title="E27">A13</font></i>: 
( <font color="Maroon" title="c5">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">R</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a>, <a class="ref" href="relat_2.html#D12" title="RELAT_2:def.12">RELAT_2:def 12</a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></i>;<br/>
<a NAME="E21:21"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:21"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E22:21"/><b>then </b><i><font color="Green" title="E28">A14</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">lensga</font>
 
<b>by </b><i><a class="txt" href="#E13:21"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E23:21"/><b>then </b><i><font color="Green" title="E29">A15</font></i>: 
<font color="Maroon" title="c6">k</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E8:21"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="#E13:21"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> $1;<br/>
<a NAME="E24:21"/><i><font color="Green" title="E30">A16</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green" title="E31">A17</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_2"/><i><font color="Green" title="E32">A18</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="#E23:21"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E3:21_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E23:21"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:21_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="#E6:21"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E5:21_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:21_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:21_2"/><i><font color="Green" title="E33">A19</font></i>: 
( <font color="Maroon" title="c15">l</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="triang_1.html#K2" title="TRIANG_1:func.2">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="triang_1.html#K2" title="TRIANG_1:func.2">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="c15">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c6">k</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">l</font> = <font color="Maroon" title="c15">l</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E8:21_2"><i><font color="Green" title="E33">A19</font></i></a></i>;<br/>
<a NAME="E10:21_2"/><i><font color="Green" title="E34">A20</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">l</font> &amp; <font color="Maroon" title="c16">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">lensgb</font> )
 
<b>by </b><i><a class="txt" href="#E15:21"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="#E8:21_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:21_2"/><i><font color="Green" title="E35">A21</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c16">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c16">l</font>
 
<b>by </b><i><a class="txt" href="#E8:21_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E12:21_2"/><i><font color="Green" title="E36">A22</font></i>: 
<font color="Maroon" title="c16">l</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
 
<b>by </b><i><a class="txt" href="#E1:21_2"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="#E8:21_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_1"/>
( <font color="Maroon" title="c16">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> or <font color="Maroon" title="c16">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> or <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">l</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1"><b>suppose </b></a><a NAME="E1:21_2_1_1"/>
<font color="Maroon" title="c16">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:21_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="#E4:21"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E2:21_2"><i><font color="Green" title="E32">A18</font></i></a>, <a class="txt" href="#E8:21_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E3:21_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E23:21"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:21_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E6:21"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_2"><b>suppose </b></a><a NAME="E1:21_2_1_2"/><i><font color="Green" title="E37">A23</font></i>: 
<font color="Maroon" title="c16">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:21_2_1_2"/><b>then </b>
<font color="Maroon" title="c16">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">lensga</font>
 <b>by </b><i><a class="txt" href="#E22:21"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:21_2_1_2"/><b>then </b><i><font color="Green" title="E38">A24</font></i>: 
<font color="Maroon" title="c16">l</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E13:21"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="#E10:21_2"><i><font color="Green" title="E34">A20</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:21_2_1_2"/>
<font color="Maroon" title="c16">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k'</font>
 <b>by </b><i><a class="txt" href="#E11:21"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="#E1:21_2_1_2"><i><font color="Green" title="E37">A23</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:21_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c16">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c6">k</font>
 <b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E4:21"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E8:21_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="#E10:21_2"><i><font color="Green" title="E34">A20</font></i></a>, <a class="txt" href="#E11:21_2"><i><font color="Green" title="E35">A21</font></i></a>, <a class="ref" href="polynom2.html#T11" target="_self" title="POLYNOM2:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E6:21_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E23:21"><i><font color="Green" title="E29">A15</font></i></a>, <a class="txt" href="#E1:21_2_1_2"><i><font color="Green" title="E37">A23</font></i></a>, <a class="txt" href="#E3:21_2_1_2"><i><font color="Green" title="E38">A24</font></i></a>, <a class="txt" href="polynom2.html#E8"><i><font color="Green" title="E6">Lm4</font></i></a>, <a class="ref" href="polynom2.html#T10" target="_self" title="POLYNOM2:th.10">Th10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_3"><b>suppose </b></a><a NAME="E1:21_2_1_3"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:21_2_1_3"/><b>then </b><i><font color="Green" title="E37">A25</font></i>: 
<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="c16">l</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:21_2_1_3"/><b>then </b><i><font color="Green" title="E38">A26</font></i>: 
<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="c16">l</font>
 <b>by </b><i><a class="txt" href="#E12:21_2"><i><font color="Green" title="E36">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:21_2_1_3"/><i><font color="Green" title="E39">A27</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E5:21_2_1_3"/><b>then </b><i><font color="Green" title="E40">A28</font></i>: 
<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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:21_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="#E2:21_2_1_3"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="polynom2.html#T4" target="_self" title="POLYNOM2:th.4">Th4</a></i>;<br/><a NAME="E6:21_2_1_3"/><b>then </b><i><font color="Green" title="E41">A29</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="p3">(<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>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c16">l</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E8:21_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="#E3:21_2_1_3"><i><font color="Green" title="E38">A26</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><a NAME="E7:21_2_1_3"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <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>
 <b>by </b><i><a class="txt" href="#E8:21_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="#E2:21_2_1_3"><i><font color="Green" title="E37">A25</font></i></a>, <a class="txt" href="#E4:21_2_1_3"><i><font color="Green" title="E39">A27</font></i></a>, <a class="ref" href="polynom2.html#T4" target="_self" title="POLYNOM2:th.4">Th4</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E8:21_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:21_2_1_3"><i><font color="Green" title="E40">A28</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:21_2_1_3"/><b>then </b><i><font color="Green" title="E42">A30</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2_1_3_1"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:21_2_1_3_1"/><b>then </b>
<font color="Maroon" title="c6">k</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="c6">k</font>
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E4:21"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E5:21_2_1_3"><i><font color="Green" title="E40">A28</font></i></a>, <a class="ref" href="polynom2.html#T10" target="_self" title="POLYNOM2:th.10">Th10</a></i>;<br/><b>hence </b><a NAME="E3:21_2_1_3_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E11:21_2_1_3"/><b>then </b>
not <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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#R2" title="HIDDEN:pred.2">in</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>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E12:21_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E9:21_2_1_3"><i><font color="Green" title="E42">A30</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E13:21_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:21"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">l'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:21_2_1_3"/><i><font color="Green" title="E43">A31</font></i>: 
( <font color="Maroon" title="c17">l'</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">l'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">l'</font> = <font color="Maroon" title="c17">l'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E15:21_2_1_3"><i><font color="Green" title="E43">A31</font></i></a></i>;<br/><a NAME="E17:21_2_1_3"/><i><font color="Green" title="E44">A32</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c18">l'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">l'</font>
 <b>by </b><i><a class="txt" href="#E15:21_2_1_3"><i><font color="Green" title="E43">A31</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E18:21_2_1_3"/><i><font color="Green" title="E45">A33</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">l'</font> &amp; <font color="Maroon" title="c18">l'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">lensga</font> )
 <b>by </b><i><a class="txt" href="#E13:21"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="#E15:21_2_1_3"><i><font color="Green" title="E43">A31</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E19:21_2_1_3"/><b>then </b>
<font color="Maroon" title="c18">l'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">lensgb</font>
 <b>by </b><i><a class="txt" href="#E18:21"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E20:21_2_1_3"/><b>then </b><i><font color="Green" title="E46">A34</font></i>: 
<font color="Maroon" title="c18">l'</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E15:21"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="#E18:21_2_1_3"><i><font color="Green" title="E45">A33</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2_1_3_2"/><i><font color="Green" title="E47">A35</font></i>: 
<font color="Maroon" title="c18">l'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:21_2_1_3_2"/><b>then </b>
<font color="Maroon" title="c18">l'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k'</font>
 <b>by </b><i><a class="txt" href="#E11:21"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:21_2_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c18">l'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c18">l'</font>
 <b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E4:21"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E18:21_2_1_3"><i><font color="Green" title="E45">A33</font></i></a>, <a class="ref" href="polynom2.html#T11" target="_self" title="POLYNOM2:th.11">Th11</a></i>;<br/><a NAME="E4:21_2_1_3_2"/><b>then </b>
<font color="Maroon" title="c18">l'</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
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E5:21_2_1_3"><i><font color="Green" title="E40">A28</font></i></a>, <a class="txt" href="#E15:21_2_1_3"><i><font color="Green" title="E43">A31</font></i></a>, <a class="txt" href="#E20:21_2_1_3"><i><font color="Green" title="E46">A34</font></i></a>, <a class="ref" href="polynom2.html#T10" target="_self" title="POLYNOM2:th.10">Th10</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>hence </b><a NAME="E5:21_2_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:21_2_1_3_2"><i><font color="Green" title="E47">A35</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E22:21_2_1_3"/><b>then </b>
<font color="Maroon" title="c18">l'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="#E1:21_2"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="#E15:21_2_1_3"><i><font color="Green" title="E43">A31</font></i></a>, <a class="txt" href="#E17:21_2_1_3"><i><font color="Green" title="E44">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E23:21_2_1_3"/><b>then </b><i><font color="Green" title="E47">A36</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c16">l</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="p3">(<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>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="#E6:21"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="#E23:21"><i><font color="Green" title="E29">A15</font></i></a>, <a class="txt" href="#E8:21_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="#E11:21_2"><i><font color="Green" title="E35">A21</font></i></a>, <a class="txt" href="#E15:21_2_1_3"><i><font color="Green" title="E43">A31</font></i></a>, <a class="txt" href="#E17:21_2_1_3"><i><font color="Green" title="E44">A32</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><a NAME="E24:21_2_1_3"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c16">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:21_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="#E11:21_2"><i><font color="Green" title="E35">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E25:21_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c16">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><a NAME="E26:21_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c16">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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>
 <b>by </b><i><a class="txt" href="#E20:21"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="#E6:21_2_1_3"><i><font color="Green" title="E41">A29</font></i></a>, <a class="txt" href="#E23:21_2_1_3"><i><font color="Green" title="E47">A36</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>;<br/><b>hence </b><a NAME="E27:21_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:21_2"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="#E8:21_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E25:21"/><i><font color="Green" title="E31">A37</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">k</font> <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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> &amp; ( for <font color="Olive" title="b2">j'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">k</font> <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="Olive" title="b1">j</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">j'</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> &amp; ( for <font color="Olive" title="b1">j'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">k</font> <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="c15">j</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j'</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:21_3"/><i><font color="Green" title="E32">A38</font></i>: 
( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">j'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c6">k</font> <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="c15">j</font> &amp;  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j'</font>] ) or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E2:21_3"/><i><font color="Green" title="E33">A39</font></i>: 
for <font color="Olive" title="b1">j'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">k</font> <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="c15">j</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j'</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<b>assume </b><a NAME="E3:21_3"/><i><font color="Green" title="E34">A40</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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"><span class="p2">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:21_3"/><i><font color="Green" title="E35">A41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E5:21_3"/><i><font color="Green" title="E36">A42</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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>
<b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="triang_1.html#T9" title="TRIANG_1:th.9">TRIANG_1:9</a></i>
<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
<b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>
<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="triang_1.html#K2" title="TRIANG_1:func.2">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
<b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="polynom2.html#E8"><i><font color="Green" title="E6">Lm4</font></i></a>, <a class="ref" href="triang_1.html#T9" title="TRIANG_1:th.9">TRIANG_1:9</a></i>
;<br/>
<a NAME="E7:21_3"/><b>then </b>
<font color="Maroon" title="c15">j</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:21_3"><i><font color="Green" title="E32">A38</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:21_3"/><b>then </b>
<font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:21_3"><i><font color="Green" title="E35">A41</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E9:21_3"/><b>then </b>
<font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">lensgb</font>
 
<b>by </b><i><a class="txt" href="#E15:21"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:21_3"/><b>then </b><i><font color="Green" title="E37">A43</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E15:21"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="#E5:21_3"><i><font color="Green" title="E36">A42</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:21_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E12:21_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E10:21_3"><i><font color="Green" title="E37">A43</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:21_3"/><b>then </b><i><font color="Green" title="E38">A44</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E14:21_3"/>
<font color="Maroon" title="c15">j</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:21_3"><i><font color="Green" title="E32">A38</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E15:21_3"/><b>then </b><i><font color="Green" title="E39">A45</font></i>: 
<font color="Maroon" title="c15">j</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="c13">lensga</font>
 
<b>by </b><i><a class="txt" href="#E13:21"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E16:21_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E17:21_3"/><b>then </b><i><font color="Green" title="E40">A46</font></i>: 
<font color="Maroon" title="c15">j</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E13:21"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="#E15:21_3"><i><font color="Green" title="E39">A45</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E18:21_3"/><b>then </b><i><font color="Green" title="E41">A47</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_3_2"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:21_3_2"/><b>then </b>
<font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E4:21"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E10:21_3"><i><font color="Green" title="E37">A43</font></i></a>, <a class="ref" href="polynom2.html#T10" target="_self" title="POLYNOM2:th.10">Th10</a></i>;<br/><b>hence </b><a NAME="E3:21_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:21_3"><i><font color="Green" title="E32">A38</font></i></a>, <a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E20:21_3"/><b>then </b>
not <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E21:21_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E13:21_3"><i><font color="Green" title="E38">A44</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E22:21_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:21"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E24:21_3"/><i><font color="Green" title="E42">A48</font></i>: 
( <font color="Maroon" title="c16">l</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">l</font> = <font color="Maroon" title="c16">l</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E24:21_3"><i><font color="Green" title="E42">A48</font></i></a></i>;<br/>
<a NAME="E26:21_3"/><i><font color="Green" title="E43">A49</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">l</font> &amp; <font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">lensga</font> )
 
<b>by </b><i><a class="txt" href="#E13:21"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="#E24:21_3"><i><font color="Green" title="E42">A48</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E27:21_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">l</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="c14">lensgb</font> )
 
<b>by </b><i><a class="txt" href="#E17:21"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E28:21_3"/><b>then </b><i><font color="Green" title="E44">A50</font></i>: 
<font color="Maroon" title="c17">l</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E15:21"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E29:21_3"/>
<font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E30:21_3"/><b>then </b><i><font color="Green" title="E45">A51</font></i>: 
<font color="Maroon" title="c17">l</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E26:21_3"><i><font color="Green" title="E43">A49</font></i></a>, <a class="txt" href="#E28:21_3"><i><font color="Green" title="E44">A50</font></i></a>, <a class="ref" href="polynom2.html#T4" target="_self" title="POLYNOM2:th.4">Th4</a></i>;<br/>
<a NAME="E31:21_3"/><i><font color="Green" title="E46">A52</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c17">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">l</font>
 
<b>by </b><i><a class="txt" href="#E24:21_3"><i><font color="Green" title="E42">A48</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E32:21_3"/><i><font color="Green" title="E47">A53</font></i>: 
<font color="Maroon" title="c17">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E3:21_3"><i><font color="Green" title="E34">A40</font></i></a>, <a class="txt" href="#E24:21_3"><i><font color="Green" title="E42">A48</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_3_3"/>
( <font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">j</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="21_3_3_1"><b>suppose </b></a><a NAME="E1:21_3_3_1"/><i><font color="Green" title="E48">A54</font></i>: 
<font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:21_3_3_1"/><b>then </b>
<font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E32:21_3"><i><font color="Green" title="E47">A53</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:21_3_3_1"/><b>then </b><i><font color="Green" title="E49">A55</font></i>: 
<font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_3_3_1_1_1"/>
( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">l</font> or <font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_3_1_1_1_1"><b>case </b></a><a NAME="E1:21_3_3_1_1_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:21_3_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c17">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c17">l</font>
 <b>by </b><i><a class="txt" href="#E2:21_3"><i><font color="Green" title="E33">A39</font></i></a>, <a class="txt" href="#E3:21_3_3_1"><i><font color="Green" title="E49">A55</font></i></a></i>;<br/><a NAME="E3:21_3_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E10:21_3"><i><font color="Green" title="E37">A43</font></i></a>, <a class="txt" href="#E24:21_3"><i><font color="Green" title="E42">A48</font></i></a>, <a class="txt" href="#E28:21_3"><i><font color="Green" title="E44">A50</font></i></a>, <a class="ref" href="polynom2.html#T10" target="_self" title="POLYNOM2:th.10">Th10</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>hence </b><a NAME="E4:21_3_3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:21_3"><i><font color="Green" title="E34">A40</font></i></a>, <a class="txt" href="#E24:21_3"><i><font color="Green" title="E42">A48</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_3_1_1_1_2"><b>case </b></a><a NAME="E1:21_3_3_1_1_1_2"/>
<font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:21_3_3_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k'</font>
 <b>by </b><i><a class="txt" href="#E11:21"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:21_3_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E50">A56</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c17">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c17">l</font>
 <b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E4:21"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E26:21_3"><i><font color="Green" title="E43">A49</font></i></a>, <a class="ref" href="polynom2.html#T11" target="_self" title="POLYNOM2:th.11">Th11</a></i>;<br/><a NAME="E4:21_3_3_1_1_1_2"/>
<font color="Maroon" title="c15">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><b>hence </b><a NAME="E5:21_3_3_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E10:21_3"><i><font color="Green" title="E37">A43</font></i></a>, <a class="txt" href="#E24:21_3"><i><font color="Green" title="E42">A48</font></i></a>, <a class="txt" href="#E30:21_3"><i><font color="Green" title="E45">A51</font></i></a>, <a class="txt" href="#E1:21_3_3_1"><i><font color="Green" title="E48">A54</font></i></a>, <a class="txt" href="#E3:21_3_3_1_1_1_2"><i><font color="Green" title="E50">A56</font></i></a>, <a class="ref" href="polynom2.html#T10" target="_self" title="POLYNOM2:th.10">Th10</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:21_3_3_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_3_2"><b>suppose </b></a><a NAME="E1:21_3_3_2"/><i><font color="Green" title="E48">A57</font></i>: 
<font color="Maroon" title="c17">l</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:21_3_3_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c15">j</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:21_3"><i><font color="Green" title="E32">A38</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:21_3_3_2"/><b>then </b>
<font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:21_3_3_2"/><b>then </b>
<font color="Maroon" title="c15">j</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:21_3_3_2"/><b>then </b><i><font color="Green" title="E49">A58</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p3">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c17">l</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="#E6:21"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="#E24:21_3"><i><font color="Green" title="E42">A48</font></i></a>, <a class="txt" href="#E1:21_3_3_2"><i><font color="Green" title="E48">A57</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><a NAME="E6:21_3_3_2"/><i><font color="Green" title="E50">A59</font></i>: 
for <font color="Olive" title="b1">i'</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 holds <br/><span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="Olive" title="b1">i'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">i'</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i'</font> &amp; <font color="Maroon" title="c18">i'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 implies <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c18">i'</font> )</span><br/>

<b>assume </b><a NAME="E1:21_3_3_2_1"/><i><font color="Green" title="E51">A60</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i'</font> &amp; <font color="Maroon" title="c18">i'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c18">i'</font></span><br/>

<b>assume </b><a NAME="E2:21_3_3_2_1"/><i><font color="Green" title="E52">A61</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c18">i'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_3_3_2_1_1"/>
( <font color="Maroon" title="c18">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c18">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:21_3_3_2_1_1_1"/>
<font color="Maroon" title="c18">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:21_3_3_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:21_3"><i><font color="Green" title="E34">A40</font></i></a>, <a class="txt" href="#E2:21_3_3_2_1"><i><font color="Green" title="E52">A61</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:21_3_3_2_1_1_2"/><i><font color="Green" title="E53">A62</font></i>: 
<font color="Maroon" title="c18">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:21_3_3_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">i'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="txt" href="#E1:21_3_3_2_1"><i><font color="Green" title="E51">A60</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:21_3_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E54">A63</font></i>: 
<font color="Maroon" title="c18">i'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E4:21_3"><i><font color="Green" title="E35">A41</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:21_3_3_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">i'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">lensga</font>
 <b>by </b><i><a class="txt" href="#E15:21_3"><i><font color="Green" title="E39">A45</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:21_3_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E55">A64</font></i>: 
<font color="Maroon" title="c18">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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E13:21"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="#E1:21_3_3_2_1"><i><font color="Green" title="E51">A60</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_3_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_3_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_3_3_2_1_1_2_1_1"/>
( <font color="Maroon" title="c18">i'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> or <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">i'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_3_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:21_3_3_2_1_1_2_1_1_1"/><i><font color="Green" title="E56">A65</font></i>: 
<font color="Maroon" title="c18">i'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_3_2_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_3_2_1_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_3_3_2_1_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c18">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> or <font color="Maroon" title="c18">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_3_2_1_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:21_3_3_2_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c18">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:21_3_3_2_1_1_2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</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>
 <b>by </b><i><a class="txt" href="#E4:21"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E17:21_3"><i><font color="Green" title="E40">A46</font></i></a>, <a class="txt" href="#E2:21_3_3_2_1"><i><font color="Green" title="E52">A61</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E3:21_3_3_2_1_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E17:21_3"><i><font color="Green" title="E40">A46</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:21_3_3_2_1_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E6:21"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_3_2_1_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:21_3_3_2_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c18">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:21_3_3_2_1_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">i'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="#E1:21_3_3_2_1_1_2_1_1_1"><i><font color="Green" title="E56">A65</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:21_3_3_2_1_1_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">i'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k'</font>
 <b>by </b><i><a class="txt" href="#E11:21"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:21_3_3_2_1_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c18">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c18">i'</font>
 <b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E4:21"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E1:21_3_3_2_1"><i><font color="Green" title="E51">A60</font></i></a>, <a class="ref" href="polynom2.html#T11" target="_self" title="POLYNOM2:th.11">Th11</a></i>;<br/><a NAME="E5:21_3_3_2_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E57">A66</font></i>: 
<font color="Maroon" title="c18">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E17:21_3"><i><font color="Green" title="E40">A46</font></i></a>, <a class="txt" href="#E2:21_3_3_2_1"><i><font color="Green" title="E52">A61</font></i></a>, <a class="txt" href="#E5:21_3_3_2_1_1_2"><i><font color="Green" title="E55">A64</font></i></a>, <a class="txt" href="polynom2.html#E8"><i><font color="Green" title="E6">Lm4</font></i></a>, <a class="ref" href="polynom2.html#T10" target="_self" title="POLYNOM2:th.10">Th10</a></i>;<br/><a NAME="E6:21_3_3_2_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c18">i'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="#E1:21_3"><i><font color="Green" title="E32">A38</font></i></a>, <a class="txt" href="#E1:21_3_3_2_1_1_2_1_1_1"><i><font color="Green" title="E56">A65</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:21_3_3_2_1_1_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:21_3_3_2_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E57">A66</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:21_3_3_2_1_1_2_1_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_3_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:21_3_3_2_1_1_2_1_1_2"/><i><font color="Green" title="E56">A67</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">i'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:21_3_3_2_1_1_2_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i'</font>
 <b>by </b><i><a class="txt" href="#E8:21"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:21_3_3_2_1_1_2_1_1_2"/><b>then </b>
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="c18">i'</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E4:21_3_3_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E57">A68</font></i>: 
<font color="Maroon" title="c18">i'</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E5:21_3_3_2_1_1_2_1_1_2"/><i><font color="Green" title="E58">A69</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">i'</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i'</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/><a NAME="E6:21_3_3_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E59">A70</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i'</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="#E1:21_3_3_2_1_1_2_1_1_2"><i><font color="Green" title="E56">A67</font></i></a>, <a class="txt" href="#E4:21_3_3_2_1_1_2_1_1_2"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:21_3_3_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c18">i'</font> <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"><font color="Maroon" title="c15">j</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
 <b>by </b><i><a class="txt" href="#E3:21_3_3_2_1_1_2"><i><font color="Green" title="E54">A63</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:21_3_3_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E60">A71</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i'</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E2:21_3"><i><font color="Green" title="E33">A39</font></i></a>, <a class="txt" href="#E2:21_3_3_2_1"><i><font color="Green" title="E52">A61</font></i></a>, <a class="txt" href="#E4:21_3_3_2_1_1_2_1_1_2"><i><font color="Green" title="E57">A68</font></i></a>, <a class="txt" href="#E5:21_3_3_2_1_1_2_1_1_2"><i><font color="Green" title="E58">A69</font></i></a>, <a class="txt" href="#E6:21_3_3_2_1_1_2_1_1_2"><i><font color="Green" title="E59">A70</font></i></a></i>;<br/><a NAME="E9:21_3_3_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i'</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="#E1:21_3_3_2_1_1_2_1_1_2"><i><font color="Green" title="E56">A67</font></i></a>, <a class="txt" href="#E4:21_3_3_2_1_1_2_1_1_2"><i><font color="Green" title="E57">A68</font></i></a>, <a class="txt" href="#E5:21_3_3_2_1_1_2_1_1_2"><i><font color="Green" title="E58">A69</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:21_3_3_2_1_1_2_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i'</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c18">i'</font> <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="c18">i'</font> )
 <b>by </b><i><a class="txt" href="#E8:21"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:21_3_3_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">i'</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:21_3_3_2_1_1_2"><i><font color="Green" title="E55">A64</font></i></a>, <a class="txt" href="#E4:21_3_3_2_1_1_2_1_1_2"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="polynom2.html#T4" target="_self" title="POLYNOM2:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E12:21_3_3_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E4:21_3"><i><font color="Green" title="E35">A41</font></i></a>, <a class="txt" href="#E17:21_3"><i><font color="Green" title="E40">A46</font></i></a>, <a class="txt" href="#E1:21_3_3_2_1_1_2"><i><font color="Green" title="E53">A62</font></i></a>, <a class="txt" href="#E5:21_3_3_2_1_1_2_1_1_2"><i><font color="Green" title="E58">A69</font></i></a>, <a class="txt" href="#E8:21_3_3_2_1_1_2_1_1_2"><i><font color="Green" title="E60">A71</font></i></a>, <a class="txt" href="polynom2.html#E8"><i><font color="Green" title="E6">Lm4</font></i></a>, <a class="ref" href="polynom2.html#T10" target="_self" title="POLYNOM2:th.10">Th10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:21_3_3_2_1_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:21_3_3_2"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E17:21_3"><i><font color="Green" title="E40">A46</font></i></a>, <a class="txt" href="#E18:21_3"><i><font color="Green" title="E41">A47</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:21_3_3_2"/><b>then </b><i><font color="Green" title="E51">A72</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="#E6:21"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><a NAME="E9:21_3_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:21_3_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">l'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:21_3_3_2"/><i><font color="Green" title="E52">A73</font></i>: 
( <font color="Maroon" title="c18">l'</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="triang_1.html#K2" title="TRIANG_1:func.2">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="triang_1.html#K2" title="TRIANG_1:func.2">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="c18">l'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">l'</font> = <font color="Maroon" title="c18">l'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E12:21_3_3_2"><i><font color="Green" title="E52">A73</font></i></a></i>;<br/><a NAME="E14:21_3_3_2"/><i><font color="Green" title="E53">A74</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c19">l'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c19">l'</font>
 <b>by </b><i><a class="txt" href="#E12:21_3_3_2"><i><font color="Green" title="E52">A73</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E15:21_3_3_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l'</font>
 <b>by </b><i><a class="txt" href="#E15:21"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="#E12:21_3_3_2"><i><font color="Green" title="E52">A73</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E16:21_3_3_2"/><b>then </b>
<font color="Maroon" title="c19">l'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="txt" href="#E6:21_3_3_2"><i><font color="Green" title="E50">A59</font></i></a>, <a class="txt" href="#E12:21_3_3_2"><i><font color="Green" title="E52">A73</font></i></a>, <a class="txt" href="#E14:21_3_3_2"><i><font color="Green" title="E53">A74</font></i></a></i>;<br/><a NAME="E17:21_3_3_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c17">l</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p3">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E10:21_3"><i><font color="Green" title="E37">A43</font></i></a>, <a class="txt" href="#E24:21_3"><i><font color="Green" title="E42">A48</font></i></a>, <a class="txt" href="#E31:21_3"><i><font color="Green" title="E46">A52</font></i></a>, <a class="txt" href="#E12:21_3_3_2"><i><font color="Green" title="E52">A73</font></i></a>, <a class="txt" href="#E14:21_3_3_2"><i><font color="Green" title="E53">A74</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><a NAME="E18:21_3_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c17">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E20:21"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="#E5:21_3_3_2"><i><font color="Green" title="E49">A58</font></i></a>, <a class="txt" href="#E8:21_3_3_2"><i><font color="Green" title="E51">A72</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></i>;<br/><b>hence </b><a NAME="E19:21_3_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:21"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E3:21"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E17:21_3"><i><font color="Green" title="E40">A46</font></i></a>, <a class="txt" href="#E24:21_3"><i><font color="Green" title="E42">A48</font></i></a>, <a class="txt" href="#E1:21_3_3_2"><i><font color="Green" title="E48">A57</font></i></a>, <a class="txt" href="polynom2.html#E8"><i><font color="Green" title="E6">Lm4</font></i></a>, <a class="ref" href="polynom2.html#T10" target="_self" title="POLYNOM2:th.10">Th10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E26:21"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">k</font> <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c2">A</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]
 
<b>from </b><i><a class="ref" href="polynom2.html#S5" target="_self" title="POLYNOM2:sch.5">POLYNOM2:sch 5</a>(<a class="txt" href="#E24:21"><i><font color="Green" title="E30">A16</font></i></a>, <a class="txt" href="#E25:21"><i><font color="Green" title="E31">A37</font></i></a>);<br/></i>
<b>hence </b><a NAME="E27:21"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">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="c7">i</font>
 <b>by </b><i><a class="txt" href="#E5:21"><i><font color="Green" title="E19">A5</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>
