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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1"/>
( <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>take </b>
 <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">X</font>
;<br/><b>thus </b><a NAME="E2:14_1_1_1_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/>( <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">m</font> &amp; <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="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">m</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="c3">R</font> ) ) )
 <b>by </b><i><a class="txt" href="triang_1.html#E1:14_1_1_1_1"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2"/><i><font color="Green" title="E3">A3</font></i>: 
not <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">X'</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">A'</font> = <font color="Maroon" title="c2">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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="c4">X'</font> <b>by </b><i><a class="txt" href="triang_1.html#E1:14_1_1_1_2"><i><font color="Green" title="E3">A3</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">R'</font> = <font color="Maroon" title="c3">R</font> as   <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <font color="Maroon" title="c4">X'</font> ;<br/><a NAME="E5:14_1_1_1_2"/>
 <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">A1</font> = <font color="Maroon" title="c5">A'</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c5">A'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">R'</font> <a href="triang_1.html#K1" title="TRIANG_1:func.1">|_2</a> <font color="Maroon" title="c5">A'</font></span>)</span> #) ;<br/><a NAME="E7:14_1_1_1_2"/><i><font color="Green" title="E4">A4</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">R'</font> <a href="triang_1.html#K1" title="TRIANG_1:func.1">|_2</a> <font color="Maroon" title="c5">A'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T100" title="ORDERS_1:th.100">ORDERS_1:100</a></i>;<br/><a NAME="E8:14_1_1_1_2"/>
<font color="Maroon" title="c3">R</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E1:14"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="orders_1.html#D8" title="ORDERS_1:def.8">ORDERS_1:def 8</a></i>;<br/><a NAME="E9:14_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">A</font> is <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>
 <b>by </b><i><a class="ref" href="orders_1.html#T185" title="ORDERS_1:th.185">ORDERS_1:185</a></i>;<br/><a NAME="E10:14_1_1_1_2"/><b>then </b><i><font color="Green" title="E5">A5</font></i>: 
<font color="Maroon" title="c3">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">A</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E7:14_1_1_1_2"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="relat_2.html#D14" title="RELAT_2:def.14">RELAT_2:def 14</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A1</font>) <b>-&gt; </b>   <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>  = <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,$1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>consider </b><font color="Maroon" title="c8">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">A1</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E12:14_1_1_1_2"/><i><font color="Green" title="E6">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A1</font> holds  <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E13:14_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">A1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1_1_1_2_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:14_1_1_1_2_1"/><i><font color="Green" title="E7">A7</font></i>: 
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</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="c11">x2</font> = <font color="Maroon" title="c10">x1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A1</font> <b>by </b><i><a class="txt" href="triang_1.html#E3:14_1_1_1_2_1"><i><font color="Green" title="E7">A7</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:14_1_1_1_2_1"/><i><font color="Green" title="E8">A8</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c11">x2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">A1</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T80" title="CARD_1:th.80">CARD_1:80</a></i>;<br/>
<a NAME="E6:14_1_1_1_2_1"/>
 <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c11">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">A1</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T62" title="ORDERS_2:th.62">ORDERS_2:62</a></i>;<br/>
<a NAME="E7:14_1_1_1_2_1"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c11">x2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">A1</font>
 
<b>by </b><i><a class="ref" href="triang_1.html#T3" target="_self" title="TRIANG_1:th.3">Th3</a></i>;<br/>
<a NAME="E8:14_1_1_1_2_1"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c11">x2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">A1</font>
 
<b>by </b><i><a class="txt" href="triang_1.html#E5:14_1_1_1_2_1"><i><font color="Green" title="E8">A8</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E9:14_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E9">A9</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c11">x2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">A1</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="E10:14_1_1_1_2_1"/><i><font color="Green" title="E10">A10</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c11">x2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="triang_1.html#E12:14_1_1_1_2"><i><font color="Green" title="E6">A6</font></i></a>, <a class="txt" href="triang_1.html#E3:14_1_1_1_2_1"><i><font color="Green" title="E7">A7</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">y1</font> = <font color="Maroon" title="c9">y</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E12:14_1_1_1_2_1"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">y1</font>
 
<b>by </b><i><a class="txt" href="triang_1.html#E10:14_1_1_1_2_1"><i><font color="Green" title="E10">A10</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E13:14_1_1_1_2_1"/>
<font color="Maroon" title="c9">y</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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">A1</font></span>)</span>
 <b>by </b><i><a class="txt" href="triang_1.html#E9:14_1_1_1_2_1"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="triang_1.html#E10:14_1_1_1_2_1"><i><font color="Green" title="E10">A10</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">f1</font> = <font color="Maroon" title="c8">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">A1</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">A1</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></i>;<br/><a NAME="E15:14_1_1_1_2"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A1</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x1</font>, <font color="Maroon" title="c11">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:14_1_1_1_2_2"/><i><font color="Green" title="E7">A11</font></i>: 
( <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A1</font> &amp; <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">x1'</font> = <font color="Maroon" title="c10">x1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A1</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">x2'</font> = <font color="Maroon" title="c11">x2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A1</font> <b>by </b><i><a class="txt" href="triang_1.html#E1:14_1_1_1_2_2"><i><font color="Green" title="E7">A11</font></i></a></i>;<br/>
<a NAME="E4:14_1_1_1_2_2"/>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1'</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="triang_1.html#E12:14_1_1_1_2"><i><font color="Green" title="E6">A6</font></i></a></i>;<br/>
<a NAME="E5:14_1_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1'</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2'</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="triang_1.html#E12:14_1_1_1_2"><i><font color="Green" title="E6">A6</font></i></a>, <a class="txt" href="triang_1.html#E1:14_1_1_1_2_2"><i><font color="Green" title="E7">A11</font></i></a></i>;<br/>
<a NAME="E6:14_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E8">A12</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2'</font></span>)</span>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1_2_2_1_1"/>
( <font color="Maroon" title="c12">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2'</font> or <font color="Maroon" title="c12">x1'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x2'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c12">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_2_1_1_2"/><i><font color="Green" title="E9">A13</font></i>: 
<font color="Maroon" title="c12">x1'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x2'</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E10">A14</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x1'</font>,<font color="Maroon" title="c13">x2'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">A</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">x2'</font>,<font color="Maroon" title="c12">x1'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E10:14_1_1_1_2"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/><div><i><font color="Green" title="E11">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_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="E1:14_1_1_1_2_2_1_1_2_1_1"/>
( <font color="Maroon" title="c12">x1'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">x2'</font> or <font color="Maroon" title="c13">x2'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">x1'</font> )
 </span><b>by </b><i><a class="txt" href="triang_1.html#E2:14_1_1_1_2_2_1_1_2"><i><font color="Green" title="E10">A14</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c12">x1'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">x2'</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">x1'</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c13">x2'</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E1:14_1_1_1_2_2_1_1_2"><i><font color="Green" title="E9">A13</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/><a NAME="E3:14_1_1_1_2_2_1_1_2_1_1_1"/><b>then </b>
 <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2'</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T64" title="ORDERS_2:th.64">ORDERS_2:64</a></i>;<br/><b>hence </b><a NAME="E4:14_1_1_1_2_2_1_1_2_1_1_1"/>
 <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2'</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E6:14_1_1_1_2_2"><i><font color="Green" title="E8">A12</font></i></a>, <a class="ref" href="triang_1.html#T3" target="_self" title="TRIANG_1:th.3">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c13">x2'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">x1'</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">x2'</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c12">x1'</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E1:14_1_1_1_2_2_1_1_2"><i><font color="Green" title="E9">A13</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/><a NAME="E3:14_1_1_1_2_2_1_1_2_1_1_2"/><b>then </b>
 <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1'</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T64" title="ORDERS_2:th.64">ORDERS_2:64</a></i>;<br/><b>hence </b><a NAME="E4:14_1_1_1_2_2_1_1_2_1_1_2"/>
 <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2'</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E6:14_1_1_1_2_2"><i><font color="Green" title="E8">A12</font></i></a>, <a class="ref" href="triang_1.html#T3" target="_self" title="TRIANG_1:th.3">Th3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_1_1_2_2_1_1_2_2"/><i><font color="Green" title="E12">A16</font></i>: 
<font color="Maroon" title="c12">x1'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">x2'</font>
 ;<br/><a NAME="E2:14_1_1_1_2_2_1_1_2_2"/><b>then </b><i><font color="Green" title="E13">A17</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x1'</font>,<font color="Maroon" title="c13">x2'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">A</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">x2'</font>,<font color="Maroon" title="c12">x1'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E10:14_1_1_1_2"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1_2_2_1_1_2_2_1"/>
( <font color="Maroon" title="c12">x1'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">x2'</font> or <font color="Maroon" title="c13">x2'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">x1'</font> )
 </span><b>by </b><i><a class="txt" href="triang_1.html#E2:14_1_1_1_2_2_1_1_2_2"><i><font color="Green" title="E13">A17</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c12">x1'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">x2'</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c12">x1'</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c13">x2'</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E1:14_1_1_1_2_2_1_1_2_2"><i><font color="Green" title="E12">A16</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/><a NAME="E3:14_1_1_1_2_2_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c12">x1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2'</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T57" title="ORDERS_2:th.57">ORDERS_2:57</a></i>;<br/><b>hence </b><a NAME="E4:14_1_1_1_2_2_1_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="triang_1.html#E3:14_1_1_1_2_2_1_1_2"><i><font color="Green" title="E11">A15</font></i></a>, <a class="ref" href="orders_2.html#T62" title="ORDERS_2:th.62">ORDERS_2:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_2_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c13">x2'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">x1'</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c13">x2'</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c12">x1'</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E1:14_1_1_1_2_2_1_1_2_2"><i><font color="Green" title="E12">A16</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/><a NAME="E3:14_1_1_1_2_2_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c13">x2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1'</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T57" title="ORDERS_2:th.57">ORDERS_2:57</a></i>;<br/><b>hence </b><a NAME="E4:14_1_1_1_2_2_1_1_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="triang_1.html#E3:14_1_1_1_2_2_1_1_2"><i><font color="Green" title="E11">A15</font></i></a>, <a class="ref" href="orders_2.html#T62" title="ORDERS_2:th.62">ORDERS_2:62</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:14_1_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:14_1_1_1_2_2"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E16:14_1_1_1_2"/><b>then </b><i><font color="Green" title="E7">A18</font></i>: 
<font color="Maroon" title="c9">f1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/><a NAME="E17:14_1_1_1_2"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">A1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">A1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>;<br/><a NAME="E18:14_1_1_1_2"/><b>then </b><i><font color="Green" title="E8">A19</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">A1</font></span>)</span>
 <b>by </b><i><a class="txt" href="triang_1.html#E16:14_1_1_1_2"><i><font color="Green" title="E7">A18</font></i></a>, <a class="ref" href="finseq_4.html#T78" title="FINSEQ_4:th.78">FINSEQ_4:78</a></i>;<br/><a NAME="E19:14_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">A1</font></span>)</span>
 <b>by </b><i><a class="txt" href="triang_1.html#E16:14_1_1_1_2"><i><font color="Green" title="E7">A18</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">g1</font> = <font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E21:14_1_1_1_2"/><i><font color="Green" title="E9">A20</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E22:14_1_1_1_2"/><b>then </b><i><font color="Green" title="E10">A21</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">A1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="triang_1.html#E16:14_1_1_1_2"><i><font color="Green" title="E7">A18</font></i></a>, <a class="txt" href="triang_1.html#E18:14_1_1_1_2"><i><font color="Green" title="E8">A19</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c10">g1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>take </b>
<font color="Maroon" title="c11">g</font>
;<br/><b>thus </b><a NAME="E24:14_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E16:14_1_1_1_2"><i><font color="Green" title="E7">A18</font></i></a>, <a class="txt" href="triang_1.html#E21:14_1_1_1_2"><i><font color="Green" title="E9">A20</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><a NAME="E25:14_1_1_1_2"/><i><font color="Green" title="E11">A22</font></i>: 
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A1</font>  st <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b2">x2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c13">x2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A1</font>;<br/>

<b>assume </b><a NAME="E1:14_1_1_1_2_3"/><i><font color="Green" title="E12">A23</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x2</font>
 ;<br/>

<a NAME="E2:14_1_1_1_2_3"/><i><font color="Green" title="E13">A24</font></i>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_1.html#D11" title="CARD_1:def.11">CARD_1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">Cx1</font> =  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1</font></span>)</span> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c15">Cx2</font> =  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2</font></span>)</span> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> ;<br/>
<a NAME="E5:14_1_1_1_2_3"/>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="triang_1.html#E12:14_1_1_1_2"><i><font color="Green" title="E6">A6</font></i></a></i>;<br/>
<a NAME="E6:14_1_1_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="triang_1.html#E12:14_1_1_1_2"><i><font color="Green" title="E6">A6</font></i></a>, <a class="txt" href="triang_1.html#E1:14_1_1_1_2_3"><i><font color="Green" title="E12">A23</font></i></a></i>;<br/>
<a NAME="E7:14_1_1_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p3">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p3">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="E8:14_1_1_1_2_3"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
;<br/>
<a NAME="E9:14_1_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c14">Cx1</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c15">Cx2</font>
 
<b>by </b><i><a class="txt" href="triang_1.html#E2:14_1_1_1_2_3"><i><font color="Green" title="E13">A24</font></i></a>, <a class="ref" href="card_1.html#T73" title="CARD_1:th.73">CARD_1:73</a></i>;<br/>
<a NAME="E10:14_1_1_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2</font></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T4" title="CARD_2:th.4">CARD_2:4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:14_1_1_1_2_3"/><i><font color="Green" title="E14">A25</font></i>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2</font></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p1">(<span class="default"><a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E13:14_1_1_1_2_3"/><i><font color="Green" title="E15">A26</font></i>: 
( <font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c13">x2</font> &amp; not <font color="Maroon" title="c16">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_2.html#K3" title="ORDERS_2:func.3">InitSegm</a> <font color="Maroon" title="c7">A1</font>,<font color="Maroon" title="c12">x1</font> )
 
<b>by </b><i><a class="txt" href="triang_1.html#E12:14_1_1_1_2_3"><i><font color="Green" title="E14">A25</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">a</font> = <font color="Maroon" title="c16">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A1</font> <b>by </b><i><a class="txt" href="triang_1.html#E12:14_1_1_1_2_3"><i><font color="Green" title="E14">A25</font></i></a></i>;<br/>
<a NAME="E15:14_1_1_1_2_3"/><i><font color="Green" title="E16">A27</font></i>: 
( <font color="Maroon" title="c17">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c13">x2</font> &amp; not <font color="Maroon" title="c17">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c12">x1</font> )
 
<b>by </b><i><a class="txt" href="triang_1.html#E13:14_1_1_1_2_3"><i><font color="Green" title="E15">A26</font></i></a>, <a class="ref" href="orders_2.html#T57" title="ORDERS_2:th.57">ORDERS_2:57</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1_2_3_1_1"/>
(  not <font color="Maroon" title="c17">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">x1</font> or <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x1</font> )
 </span><b>by </b><i><a class="txt" href="triang_1.html#E15:14_1_1_1_2_3"><i><font color="Green" title="E16">A27</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_3_1_1_1"/>
not <font color="Maroon" title="c17">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">x1</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green" title="E17">A28</font></i>: 
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c12">x1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_2_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_3_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="E1:14_1_1_1_2_3_1_1_1_1_1"/>
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> or <font color="Maroon" title="c12">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_1_1_2_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">x1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c13">x2</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E13:14_1_1_1_2_3"><i><font color="Green" title="E15">A26</font></i></a>, <a class="ref" href="orders_2.html#T57" title="ORDERS_2:th.57">ORDERS_2:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">a</font>
 ;<br/><div class="add"><a NAME="E2:14_1_1_1_2_3_1_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x1</font>,<font color="Maroon" title="c17">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E10:14_1_1_1_2"><i><font color="Green" title="E5">A5</font></i></a>, <a class="txt" href="triang_1.html#E2:14_1_1_1_2_3_1_1_1"><i><font color="Green" title="E17">A28</font></i></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/><a NAME="E3:14_1_1_1_2_3_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">x1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">a</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><b>hence </b><a NAME="E4:14_1_1_1_2_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">x1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c13">x2</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E15:14_1_1_1_2_3"><i><font color="Green" title="E16">A27</font></i></a>, <a class="ref" href="orders_2.html#T32" title="ORDERS_2:th.32">ORDERS_2:32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:14_1_1_1_2_3_1_1_1"/>
<font color="Maroon" title="c12">x1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c13">x2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_2_3_1_1_2"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_1_1_2_3_1_1_2"/>
<font color="Maroon" title="c12">x1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c13">x2</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E13:14_1_1_1_2_3"><i><font color="Green" title="E15">A26</font></i></a>, <a class="ref" href="orders_2.html#T57" title="ORDERS_2:th.57">ORDERS_2:57</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:14_1_1_1_2_3"/>
<font color="Maroon" title="c12">x1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c13">x2</font>
 ;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon" title="c12">n</font>, <font color="Maroon" title="c13">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E26:14_1_1_1_2"/><i><font color="Green" title="E12">A29</font></i>: 
( <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">g</font> &amp; <font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">g</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</font> )
 ;<br/><a NAME="E27:14_1_1_1_2"/><b>then </b><i><font color="Green" title="E13">A30</font></i>: 
( <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E16:14_1_1_1_2"><i><font color="Green" title="E7">A18</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">gn</font> = <font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A1</font> <b>by </b><i><a class="txt" href="triang_1.html#E22:14_1_1_1_2"><i><font color="Green" title="E10">A21</font></i></a>, <a class="txt" href="triang_1.html#E26:14_1_1_1_2"><i><font color="Green" title="E12">A29</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">gm</font> = <font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">A1</font> <b>by </b><i><a class="txt" href="triang_1.html#E22:14_1_1_1_2"><i><font color="Green" title="E10">A21</font></i></a>, <a class="txt" href="triang_1.html#E26:14_1_1_1_2"><i><font color="Green" title="E12">A29</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="E30:14_1_1_1_2"/><i><font color="Green" title="E14">A31</font></i>: 
( <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">gn</font> &amp; <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">gm</font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E16:14_1_1_1_2"><i><font color="Green" title="E7">A18</font></i></a>, <a class="txt" href="triang_1.html#E27:14_1_1_1_2"><i><font color="Green" title="E13">A30</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/><a NAME="E31:14_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c14">gn</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c15">gm</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E25:14_1_1_1_2"><i><font color="Green" title="E11">A22</font></i></a>, <a class="txt" href="triang_1.html#E26:14_1_1_1_2"><i><font color="Green" title="E12">A29</font></i></a></i>;<br/><a NAME="E32:14_1_1_1_2"/><b>then </b><i><font color="Green" title="E15">A32</font></i>: 
( <font color="Maroon" title="c14">gn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">gm</font> &amp; <font color="Maroon" title="c14">gn</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">gm</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/><a NAME="E33:14_1_1_1_2"/><i><font color="Green" title="E16">A33</font></i>: 
<font color="Maroon" title="c3">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">R</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E34:14_1_1_1_2"/><i><font color="Green" title="E17">A34</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">gn</font>,<font color="Maroon" title="c15">gm</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="triang_1.html#E32:14_1_1_1_2"><i><font color="Green" title="E15">A32</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E35:14_1_1_1_2"/>
( <font color="Maroon" title="c14">gn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c15">gm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">m</font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E26:14_1_1_1_2"><i><font color="Green" title="E12">A29</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E36:14_1_1_1_2"/>
( <font color="Maroon" title="c11">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">m</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">m</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="c3">R</font> )
 <b>by </b><i><a class="txt" href="triang_1.html#E26:14_1_1_1_2"><i><font color="Green" title="E12">A29</font></i></a>, <a class="txt" href="triang_1.html#E30:14_1_1_1_2"><i><font color="Green" title="E14">A31</font></i></a>, <a class="txt" href="triang_1.html#E33:14_1_1_1_2"><i><font color="Green" title="E16">A33</font></i></a>, <a class="txt" href="triang_1.html#E34:14_1_1_1_2"><i><font color="Green" title="E17">A34</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
