<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">Y</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">PA</font>, <font color="Olive" title="b2">PB</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font> holds   <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Olive" title="b2">PB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Olive" title="b1">PA</font></span>)</span> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Olive" title="b2">PB</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">PA</font>, <font color="Maroon" title="c3">PB</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span></span><br/>



<a NAME="E1:35"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c2">PA</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#T19" target="_self" title="PARTIT1:th.19">Th19</a></span>;<br/>
<a NAME="E2:35"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c3">PB</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#T19" target="_self" title="PARTIT1:th.19">Th19</a></span>;<br/>
<a NAME="E3:35"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">PA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/>
<a NAME="E4:35"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">PB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/>
<a NAME="E5:35"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
 <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">p0</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><span class="kw"> such that </span><br/><a NAME="E3:35_1"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<font color="Maroon" title="c6">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">p0</font>
 <span class="kw">and </span><br/><a NAME="E5:35_1"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">p0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="partit1.html#D6" target="_self" title="PARTIT1:def.6">Def6</a></span>;<br/>
<a NAME="E6:35_1"/><span class="lab"><font color="Green" title="E34">A10</font></span>: 
<font color="Maroon" title="c6">p0</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c2">PA</font>,<font color="Maroon" title="c3">PB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="partit1.html#D5" target="_self" title="PARTIT1:def.5">Def5</a></span>;<br/>
<a NAME="E7:35_1"/><span class="lab"><font color="Green" title="E35">A11</font></span>: 
<font color="Maroon" title="c6">p0</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c2">PA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/>
<a NAME="E8:35_1"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
<font color="Maroon" title="c6">p0</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c3">PB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">A1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:35_1"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
<font color="Maroon" title="c7">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PA</font>
 <span class="kw">and </span><br/><a NAME="E11:35_1"/>
<font color="Maroon" title="c7">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E12:35_1"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
<font color="Maroon" title="c6">p0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c7">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="partit1.html#D1" target="_self" title="PARTIT1:def.1">Def1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:35_1"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">and </span><br/><a NAME="E15:35_1"/><span class="lab"><font color="Green" title="E40">A16</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E12:35_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">Ca</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">PA</font> :  ex <font color="Olive" title="b2">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Y</font> st <br/>( 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> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font> holds <br/> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font>, <font color="Olive" title="b6">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b5">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">p1</font> &amp; <font color="Olive" title="b6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b5">p2</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">p2</font> ) ) ) </span>}</span>  as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">pb</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">Ca</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">Cb</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">PB</font> :  ex <font color="Olive" title="b2">q</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ca</font> &amp; <font color="Olive" title="b1">p</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span>}</span>  as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">x9</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">fx</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c12">x9</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Y</font> ;<br/>
<a NAME="E21:35_1"/><span class="lab"><font color="Green" title="E41">A17</font></span>: 
<font color="Maroon" title="c13">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/>
<a NAME="E22:35_1"/><span class="lab"><font color="Green" title="E42">A18</font></span>: 
<font color="Maroon" title="c13">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">fx</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="txt" href="#E21:35_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/>
<a NAME="E23:35_1"/><span class="lab"><font color="Green" title="E43">A19</font></span>: 
( 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> <font color="Maroon" title="c13">fx</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">fx</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c13">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> &amp; <font color="Olive" title="b4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Maroon" title="c13">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">p2</font> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/>
<a NAME="E24:35_1"/><span class="lab"><font color="Green" title="E44">A20</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ca</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E15:35_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E21:35_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E22:35_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E23:35_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">y5</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E26:35_1"/><span class="lab"><font color="Green" title="E45">A21</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">y5</font>
 <span class="kw">and </span><br/><a NAME="E27:35_1"/><span class="lab"><font color="Green" title="E46">A22</font></span>: 
<font color="Maroon" title="c14">y5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ca</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="txt" href="#E24:35_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">z5</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E29:35_1"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
<font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">z5</font>
 <span class="kw">and </span><br/><a NAME="E30:35_1"/><span class="lab"><font color="Green" title="E48">A24</font></span>: 
<font color="Maroon" title="c15">z5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E31:35_1"/><span class="lab"><font color="Green" title="E49">A25</font></span>: 
<font color="Maroon" title="c14">y5</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c15">z5</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E29:35_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E32:35_1"/><span class="lab"><font color="Green" title="E50">A26</font></span>: 
<font color="Maroon" title="c15">z5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Cb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:35_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E30:35_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E31:35_1"><span class="lab"><font color="Green" title="E49">A25</font></span></a></span>;<br/>
<a NAME="E33:35_1"/><span class="lab"><font color="Green" title="E51">A27</font></span>: 
<font color="Maroon" title="c9">Ca</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ca</font> or <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_1"/><span class="lab"><font color="Green" title="E52">A28</font></span>: 
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ca</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font></span><br/>

<a NAME="E2:35_1_1"/><span class="lab"><font color="Green" title="E53">A29</font></span>: 
 ex <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">PA</font> st <br/>( <font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  ex <font color="Olive" title="b2">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Y</font> st <br/>( 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> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font> holds <br/> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font>, <font color="Olive" title="b6">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b5">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">p1</font> &amp; <font color="Olive" title="b6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b5">p2</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">p2</font> ) ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E52">A28</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:35_1_1"/>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E53">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E34:35_1"/><span class="lab"><font color="Green" title="E52">A30</font></span>: 
<font color="Maroon" title="c10">pb</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c2">PA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:35_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E33:35_1"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="ref" href="partit1.html#D1" target="_self" title="PARTIT1:def.1">Def1</a></span>;<br/>
<a NAME="E35:35_1"/><span class="lab"><font color="Green" title="E53">A31</font></span>: 
<font color="Maroon" title="c10">pb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c11">Cb</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">pb</font> or <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c11">Cb</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_2"/><span class="lab"><font color="Green" title="E54">A32</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">pb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c11">Cb</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c17">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_2"/><span class="lab"><font color="Green" title="E55">A33</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">y</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_2"/><span class="lab"><font color="Green" title="E56">A34</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ca</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_2"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:35_1_2"/><span class="lab"><font color="Green" title="E57">A35</font></span>: 
 ex <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">PA</font> st <br/>( <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  ex <font color="Olive" title="b2">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Y</font> st <br/>( 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> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font> holds <br/> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font>, <font color="Olive" title="b6">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b5">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">p1</font> &amp; <font color="Olive" title="b6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b5">p2</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">p2</font> ) ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_2"><span class="lab"><font color="Green" title="E56">A34</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:35_1_2"/><span class="lab"><font color="Green" title="E58">A36</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">z</font>
 <span class="kw">and </span><br/><a NAME="E8:35_1_2"/><span class="lab"><font color="Green" title="E59">A37</font></span>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E3:35_1_2"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E5:35_1_2"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E9:35_1_2"/><span class="lab"><font color="Green" title="E60">A38</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_2"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E7:35_1_2"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E10:35_1_2"/><span class="lab"><font color="Green" title="E61">A39</font></span>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Cb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_2"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="txt" href="#E8:35_1_2"><span class="lab"><font color="Green" title="E59">A37</font></span></a>, <a class="txt" href="#E9:35_1_2"><span class="lab"><font color="Green" title="E60">A38</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E11:35_1_2"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c11">Cb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_2"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="txt" href="#E10:35_1_2"><span class="lab"><font color="Green" title="E61">A39</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E36:35_1"/><span class="lab"><font color="Green" title="E54">A40</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c11">Cb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">pb</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c11">Cb</font> or <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">pb</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_3"/><span class="lab"><font color="Green" title="E55">A41</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c11">Cb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">pb</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c17">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_3"/><span class="lab"><font color="Green" title="E56">A42</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">y1</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_3"/><span class="lab"><font color="Green" title="E57">A43</font></span>: 
<font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Cb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_3"><span class="lab"><font color="Green" title="E55">A41</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:35_1_3"/><span class="lab"><font color="Green" title="E58">A44</font></span>: 
 ex <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">PB</font> st <br/>( <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  ex <font color="Olive" title="b2">q</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ca</font> &amp; <font color="Olive" title="b1">p</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_3"><span class="lab"><font color="Green" title="E57">A43</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:35_1_3"/><span class="lab"><font color="Green" title="E59">A45</font></span>: 
<font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ca</font>
 <span class="kw">and </span><br/><a NAME="E8:35_1_3"/><span class="lab"><font color="Green" title="E60">A46</font></span>: 
<font color="Maroon" title="c17">y1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_3"><span class="lab"><font color="Green" title="E58">A44</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">pd</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:35_1_3"/><span class="lab"><font color="Green" title="E61">A47</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">pd</font>
 <span class="kw">and </span><br/><a NAME="E11:35_1_3"/><span class="lab"><font color="Green" title="E62">A48</font></span>: 
<font color="Maroon" title="c19">pd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E3:35_1_3"><span class="lab"><font color="Green" title="E56">A42</font></span></a>, <a class="txt" href="#E5:35_1_3"><span class="lab"><font color="Green" title="E58">A44</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E12:35_1_3"/><span class="lab"><font color="Green" title="E63">A49</font></span>: 
 ex <font color="Olive" title="b1">pp</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">PA</font> st <br/>( <font color="Maroon" title="c18">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">pp</font> &amp;  ex <font color="Olive" title="b2">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Y</font> st <br/>( 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> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">pp</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font> holds <br/> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font>, <font color="Olive" title="b6">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b5">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">p1</font> &amp; <font color="Olive" title="b6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b5">p2</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">p2</font> ) ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_3"><span class="lab"><font color="Green" title="E59">A45</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">fd</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Y</font><span class="kw"> such that </span><br/><a NAME="E14:35_1_3"/><span class="lab"><font color="Green" title="E64">A50</font></span>: 
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> <font color="Maroon" title="c20">fd</font>
 <span class="kw">and </span><br/><a NAME="E15:35_1_3"/><span class="lab"><font color="Green" title="E65">A51</font></span>: 
<font color="Maroon" title="c20">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E16:35_1_3"/><span class="lab"><font color="Green" title="E66">A52</font></span>: 
<font color="Maroon" title="c20">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">fd</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">q</font>
 <span class="kw">and </span><br/><a NAME="E17:35_1_3"/><span class="lab"><font color="Green" title="E67">A53</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">fd</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c20">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> &amp; <font color="Olive" title="b4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Maroon" title="c20">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:35_1_3"><span class="lab"><font color="Green" title="E63">A49</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">x1</font> = <font color="Maroon" title="c16">x1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_3"><span class="lab"><font color="Green" title="E56">A42</font></span></a>, <a class="txt" href="#E5:35_1_3"><span class="lab"><font color="Green" title="E58">A44</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">f</font> = <font color="Maroon" title="c20">fd</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Y</font> ;<br/>
<a NAME="E20:35_1_3"/><span class="lab"><font color="Green" title="E68">A54</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">fd</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/>
<a NAME="E21:35_1_3"/><span class="lab"><font color="Green" title="E69">A55</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">fd</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:35_1_3"><span class="lab"><font color="Green" title="E68">A54</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/>
<a NAME="E22:35_1_3"/><span class="lab"><font color="Green" title="E70">A56</font></span>: 
1 <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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">fd</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_3"><span class="lab"><font color="Green" title="E64">A50</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E23:35_1_3"/><span class="lab"><font color="Green" title="E71">A57</font></span>: 
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> <font color="Maroon" title="c22">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_3"><span class="lab"><font color="Green" title="E69">A55</font></span></a>, <a class="txt" href="#E22:35_1_3"><span class="lab"><font color="Green" title="E70">A56</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E24:35_1_3"/><span class="lab"><font color="Green" title="E72">A58</font></span>: 
<font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">fd</font></span>)</span> <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="c17">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_3"><span class="lab"><font color="Green" title="E56">A42</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E25:35_1_3"/><span class="lab"><font color="Green" title="E73">A59</font></span>: 
<font color="Maroon" title="c17">y1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c18">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_3"><span class="lab"><font color="Green" title="E60">A46</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c23">z0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E27:35_1_3"/><span class="lab"><font color="Green" title="E74">A60</font></span>: 
( <font color="Maroon" title="c23">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">y1</font> &amp; <font color="Maroon" title="c23">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:35_1_3"><span class="lab"><font color="Green" title="E73">A59</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E28:35_1_3"/><span class="lab"><font color="Green" title="E75">A61</font></span>: 
<font color="Maroon" title="c23">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">y1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:35_1_3"><span class="lab"><font color="Green" title="E74">A60</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E29:35_1_3"/><span class="lab"><font color="Green" title="E76">A62</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">fd</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">fd</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E30:35_1_3"/><span class="lab"><font color="Green" title="E77">A63</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">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> <font color="Maroon" title="c20">fd</font> holds <br/><font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font> &amp; <font color="Maroon" title="c24">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> <font color="Maroon" title="c20">fd</font> implies <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_3_1"/><span class="lab"><font color="Green" title="E78">A64</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">k</font> &amp; <font color="Maroon" title="c24">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> <font color="Maroon" title="c20">fd</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span><br/>

<a NAME="E2:35_1_3_1"/><span class="lab"><font color="Green" title="E79">A65</font></span>: 
<font color="Maroon" title="c24">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> <font color="Maroon" title="c20">fd</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:35_1_3"><span class="lab"><font color="Green" title="E76">A62</font></span></a>, <a class="txt" href="#E1:35_1_3_1"><span class="lab"><font color="Green" title="E78">A64</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:35_1_3_1"/>
<font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_3_1"><span class="lab"><font color="Green" title="E79">A65</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E31:35_1_3"/><span class="lab"><font color="Green" title="E78">A66</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">fd</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c21">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">fd</font></span>)</span> <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="c21">x1</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_3"><span class="lab"><font color="Green" title="E64">A50</font></span></a>, <a class="txt" href="#E15:35_1_3"><span class="lab"><font color="Green" title="E65">A51</font></span></a>, <a class="txt" href="#E30:35_1_3"><span class="lab"><font color="Green" title="E77">A63</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E32:35_1_3"/><span class="lab"><font color="Green" title="E79">A67</font></span>: 
<font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">fd</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:35_1_3"><span class="lab"><font color="Green" title="E64">A50</font></span></a>, <a class="txt" href="#E16:35_1_3"><span class="lab"><font color="Green" title="E66">A52</font></span></a>, <a class="txt" href="#E30:35_1_3"><span class="lab"><font color="Green" title="E77">A63</font></span></a></span>;<br/>
<a NAME="E33:35_1_3"/><span class="lab"><font color="Green" title="E80">A68</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">fd</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> &amp; <font color="Olive" title="b4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">p2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">i</font> &amp; <font color="Maroon" title="c24">i</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> <font color="Maroon" title="c20">fd</font> implies  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</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="Olive" title="b2">p2</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_3_2"/><span class="lab"><font color="Green" title="E81">A69</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">i</font> &amp; <font color="Maroon" title="c24">i</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> <font color="Maroon" title="c20">fd</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</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="Olive" title="b2">p2</font> )</span><br/>

<a NAME="E2:35_1_3_2"/><span class="lab"><font color="Green" title="E82">A70</font></span>: 
<font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:35_1_3"><span class="lab"><font color="Green" title="E77">A63</font></span></a>, <a class="txt" href="#E1:35_1_3_2"><span class="lab"><font color="Green" title="E81">A69</font></span></a></span>;<br/>
<a NAME="E3:35_1_3_2"/><span class="lab"><font color="Green" title="E83">A71</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">i</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> <font color="Maroon" title="c20">fd</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_3_2"><span class="lab"><font color="Green" title="E81">A69</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E4:35_1_3_2"/><span class="lab"><font color="Green" title="E84">A72</font></span>: 
<font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">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> <font color="Maroon" title="c20">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:35_1_3"><span class="lab"><font color="Green" title="E77">A63</font></span></a>, <a class="txt" href="#E3:35_1_3_2"><span class="lab"><font color="Green" title="E83">A71</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:35_1_3_2"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</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="Olive" title="b2">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_3"><span class="lab"><font color="Green" title="E67">A53</font></span></a>, <a class="txt" href="#E1:35_1_3_2"><span class="lab"><font color="Green" title="E81">A69</font></span></a>, <a class="txt" href="#E2:35_1_3_2"><span class="lab"><font color="Green" title="E82">A70</font></span></a>, <a class="txt" href="#E4:35_1_3_2"><span class="lab"><font color="Green" title="E84">A72</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E34:35_1_3"/><span class="lab"><font color="Green" title="E81">A73</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">f</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> &amp; <font color="Olive" title="b4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">p2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">i</font> &amp; <font color="Maroon" title="c24">i</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> <font color="Maroon" title="c22">f</font> implies  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</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="Olive" title="b2">p2</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_3_3"/><span class="lab"><font color="Green" title="E82">A74</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">i</font>
 <span class="kw">and </span><br/><a NAME="E2:35_1_3_3"/><span class="lab"><font color="Green" title="E83">A75</font></span>: 
<font color="Maroon" title="c24">i</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> <font color="Maroon" title="c22">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</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="Olive" title="b2">p2</font> )</span><br/>

<a NAME="E3:35_1_3_3"/><span class="lab"><font color="Green" title="E84">A76</font></span>: 
<font color="Maroon" title="c24">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">fd</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_3"><span class="lab"><font color="Green" title="E69">A55</font></span></a>, <a class="txt" href="#E2:35_1_3_3"><span class="lab"><font color="Green" title="E83">A75</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E85">A77</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_3_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_3_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_3_3_1_1"/>
( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">i</font> &amp; <font color="Maroon" title="c24">i</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> <font color="Maroon" title="c20">fd</font> ) or ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">i</font> &amp; <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">fd</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_3_3"><span class="lab"><font color="Green" title="E82">A74</font></span></a>, <a class="txt" href="#E3:35_1_3_3"><span class="lab"><font color="Green" title="E84">A76</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_3_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_3_3_1_1_1"/><span class="lab"><font color="Green" title="E86">A78</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">i</font> &amp; <font color="Maroon" title="c24">i</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> <font color="Maroon" title="c20">fd</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</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="Olive" title="b2">p2</font> )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:35_1_3_3_1_1_1"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</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="Olive" title="b2">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:35_1_3"><span class="lab"><font color="Green" title="E80">A68</font></span></a>, <a class="txt" href="#E1:35_1_3_3_1_1_1"><span class="lab"><font color="Green" title="E86">A78</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_3_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_3_3_1_1_2"/><span class="lab"><font color="Green" title="E86">A79</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">i</font> &amp; <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">fd</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</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="Olive" title="b2">p2</font> )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:35_1_3_3_1_1_2"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</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="Olive" title="b2">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_3"><span class="lab"><font color="Green" title="E58">A44</font></span></a>, <a class="txt" href="#E12:35_1_3"><span class="lab"><font color="Green" title="E63">A49</font></span></a>, <a class="txt" href="#E24:35_1_3"><span class="lab"><font color="Green" title="E72">A58</font></span></a>, <a class="txt" href="#E28:35_1_3"><span class="lab"><font color="Green" title="E75">A61</font></span></a>, <a class="txt" href="#E32:35_1_3"><span class="lab"><font color="Green" title="E79">A67</font></span></a>, <a class="txt" href="#E1:35_1_3_3_1_1_2"><span class="lab"><font color="Green" title="E86">A79</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E5:35_1_3_3"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c22">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</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="Olive" title="b2">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_3_3"><span class="lab"><font color="Green" title="E85">A77</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E35:35_1_3"/><span class="lab"><font color="Green" title="E82">A80</font></span>: 
<font color="Maroon" title="c19">pd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ca</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1_3"><span class="lab"><font color="Green" title="E61">A47</font></span></a>, <a class="txt" href="#E11:35_1_3"><span class="lab"><font color="Green" title="E62">A48</font></span></a>, <a class="txt" href="#E21:35_1_3"><span class="lab"><font color="Green" title="E69">A55</font></span></a>, <a class="txt" href="#E23:35_1_3"><span class="lab"><font color="Green" title="E71">A57</font></span></a>, <a class="txt" href="#E31:35_1_3"><span class="lab"><font color="Green" title="E78">A66</font></span></a>, <a class="txt" href="#E34:35_1_3"><span class="lab"><font color="Green" title="E81">A73</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E36:35_1_3"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">pb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35_1_3"><span class="lab"><font color="Green" title="E61">A47</font></span></a>, <a class="txt" href="#E35:35_1_3"><span class="lab"><font color="Green" title="E82">A80</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E37:35_1"/><span class="lab"><font color="Green" title="E55">A81</font></span>: 
<font color="Maroon" title="c10">pb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c11">Cb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:35_1"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="txt" href="#E36:35_1"><span class="lab"><font color="Green" title="E54">A40</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E38:35_1"/><span class="lab"><font color="Green" title="E56">A82</font></span>: 
<font color="Maroon" title="c11">Cb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">PB</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Cb</font> or <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_4"/><span class="lab"><font color="Green" title="E57">A83</font></span>: 
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Cb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font></span><br/>

<a NAME="E2:35_1_4"/><span class="lab"><font color="Green" title="E58">A84</font></span>: 
 ex <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">PB</font> st <br/>( <font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  ex <font color="Olive" title="b2">q</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ca</font> &amp; <font color="Olive" title="b1">p</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_4"><span class="lab"><font color="Green" title="E57">A83</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:35_1_4"/>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_4"><span class="lab"><font color="Green" title="E58">A84</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E39:35_1"/><span class="lab"><font color="Green" title="E57">A85</font></span>: 
<font color="Maroon" title="c10">pb</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c3">PB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:35_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E37:35_1"><span class="lab"><font color="Green" title="E55">A81</font></span></a>, <a class="txt" href="#E38:35_1"><span class="lab"><font color="Green" title="E56">A82</font></span></a>, <a class="ref" href="partit1.html#D1" target="_self" title="PARTIT1:def.1">Def1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E58">A86</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_5"/><span class="lab"><font color="Green" title="E59">A87</font></span>: 
not <font color="Maroon" title="c10">pb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">p0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_5"/><span class="lab"><font color="Green" title="E60">A88</font></span>: 
<font color="Maroon" title="c10">pb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">p0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_5"><span class="lab"><font color="Green" title="E59">A87</font></span></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:35_1_5"/><span class="lab"><font color="Green" title="E61">A89</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">pb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c6">p0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_5"><span class="lab"><font color="Green" title="E60">A88</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E5:35_1_5"/><span class="lab"><font color="Green" title="E62">A90</font></span>: 
not <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">p0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_5"><span class="lab"><font color="Green" title="E61">A89</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:35_1_5"/><span class="lab"><font color="Green" title="E63">A91</font></span>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">y1</font>
 <span class="kw">and </span><br/><a NAME="E8:35_1_5"/><span class="lab"><font color="Green" title="E64">A92</font></span>: 
<font color="Maroon" title="c17">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Cb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:35_1"><span class="lab"><font color="Green" title="E55">A81</font></span></a>, <a class="txt" href="#E4:35_1_5"><span class="lab"><font color="Green" title="E61">A89</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><a NAME="E9:35_1_5"/><span class="lab"><font color="Green" title="E65">A93</font></span>: 
 ex <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">PB</font> st <br/>( <font color="Maroon" title="c17">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  ex <font color="Olive" title="b2">q</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ca</font> &amp; <font color="Olive" title="b1">p</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1_5"><span class="lab"><font color="Green" title="E64">A92</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:35_1_5"/><span class="lab"><font color="Green" title="E66">A94</font></span>: 
<font color="Maroon" title="c18">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ca</font>
 <span class="kw">and </span><br/><a NAME="E12:35_1_5"/><span class="lab"><font color="Green" title="E67">A95</font></span>: 
<font color="Maroon" title="c17">y1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_5"><span class="lab"><font color="Green" title="E65">A93</font></span></a></span>;<br/><a NAME="E13:35_1_5"/><span class="lab"><font color="Green" title="E68">A96</font></span>: 
 ex <font color="Olive" title="b1">pp</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">PA</font> st <br/>( <font color="Maroon" title="c18">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">pp</font> &amp;  ex <font color="Olive" title="b2">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Y</font> st <br/>( 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> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">pp</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font> holds <br/> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font>, <font color="Olive" title="b6">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b5">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">p1</font> &amp; <font color="Olive" title="b6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b5">p2</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">p2</font> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_5"><span class="lab"><font color="Green" title="E66">A94</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c19">fd</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Y</font><span class="kw"> such that </span><br/><a NAME="E15:35_1_5"/><span class="lab"><font color="Green" title="E69">A97</font></span>: 
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> <font color="Maroon" title="c19">fd</font>
 <span class="kw">and </span><br/><a NAME="E16:35_1_5"/><span class="lab"><font color="Green" title="E70">A98</font></span>: 
<font color="Maroon" title="c19">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E17:35_1_5"/><span class="lab"><font color="Green" title="E71">A99</font></span>: 
<font color="Maroon" title="c19">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fd</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">q</font>
 <span class="kw">and </span><br/><a NAME="E18:35_1_5"/><span class="lab"><font color="Green" title="E72">A100</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fd</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c19">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> &amp; <font color="Olive" title="b4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Maroon" title="c19">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_5"><span class="lab"><font color="Green" title="E68">A96</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">x1</font> = <font color="Maroon" title="c16">x1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_5"><span class="lab"><font color="Green" title="E63">A91</font></span></a>, <a class="txt" href="#E9:35_1_5"><span class="lab"><font color="Green" title="E65">A93</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">f</font> = <font color="Maroon" title="c19">fd</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c20">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Y</font> ;<br/><a NAME="E21:35_1_5"/><span class="lab"><font color="Green" title="E73">A101</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fd</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c20">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><a NAME="E22:35_1_5"/><span class="lab"><font color="Green" title="E74">A102</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fd</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1_5"><span class="lab"><font color="Green" title="E73">A101</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/><a NAME="E23:35_1_5"/><span class="lab"><font color="Green" title="E75">A103</font></span>: 
1 <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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fd</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_5"><span class="lab"><font color="Green" title="E69">A97</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E24:35_1_5"/><span class="lab"><font color="Green" title="E76">A104</font></span>: 
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> <font color="Maroon" title="c21">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_5"><span class="lab"><font color="Green" title="E74">A102</font></span></a>, <a class="txt" href="#E23:35_1_5"><span class="lab"><font color="Green" title="E75">A103</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E25:35_1_5"/><span class="lab"><font color="Green" title="E77">A105</font></span>: 
<font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fd</font></span>)</span> <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="c17">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_5"><span class="lab"><font color="Green" title="E63">A91</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/><a NAME="E26:35_1_5"/><span class="lab"><font color="Green" title="E78">A106</font></span>: 
<font color="Maroon" title="c17">y1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c18">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:35_1_5"><span class="lab"><font color="Green" title="E67">A95</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c22">z0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E28:35_1_5"/><span class="lab"><font color="Green" title="E79">A107</font></span>: 
( <font color="Maroon" title="c22">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">y1</font> &amp; <font color="Maroon" title="c22">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:35_1_5"><span class="lab"><font color="Green" title="E78">A106</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E29:35_1_5"/><span class="lab"><font color="Green" title="E80">A108</font></span>: 
<font color="Maroon" title="c22">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">y1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:35_1_5"><span class="lab"><font color="Green" title="E79">A107</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E30:35_1_5"/><span class="lab"><font color="Green" title="E81">A109</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">fd</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fd</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E31:35_1_5"/><span class="lab"><font color="Green" title="E82">A110</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">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> <font color="Maroon" title="c19">fd</font> holds <br/><font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font> &amp; <font color="Maroon" title="c23">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> <font color="Maroon" title="c19">fd</font> implies <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_5_1"/><span class="lab"><font color="Green" title="E83">A111</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">k</font> &amp; <font color="Maroon" title="c23">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> <font color="Maroon" title="c19">fd</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font></span><br/>

<a NAME="E2:35_1_5_1"/><span class="lab"><font color="Green" title="E84">A112</font></span>: 
<font color="Maroon" title="c23">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> <font color="Maroon" title="c19">fd</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:35_1_5"><span class="lab"><font color="Green" title="E81">A109</font></span></a>, <a class="txt" href="#E1:35_1_5_1"><span class="lab"><font color="Green" title="E83">A111</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:35_1_5_1"/>
<font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_5_1"><span class="lab"><font color="Green" title="E84">A112</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E32:35_1_5"/><span class="lab"><font color="Green" title="E83">A113</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c19">fd</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c20">x1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fd</font></span>)</span> <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="c20">x1</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_5"><span class="lab"><font color="Green" title="E69">A97</font></span></a>, <a class="txt" href="#E16:35_1_5"><span class="lab"><font color="Green" title="E70">A98</font></span></a>, <a class="txt" href="#E31:35_1_5"><span class="lab"><font color="Green" title="E82">A110</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/><a NAME="E33:35_1_5"/><span class="lab"><font color="Green" title="E84">A114</font></span>: 
<font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fd</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:35_1_5"><span class="lab"><font color="Green" title="E69">A97</font></span></a>, <a class="txt" href="#E17:35_1_5"><span class="lab"><font color="Green" title="E71">A99</font></span></a>, <a class="txt" href="#E31:35_1_5"><span class="lab"><font color="Green" title="E82">A110</font></span></a></span>;<br/><a NAME="E34:35_1_5"/><span class="lab"><font color="Green" title="E85">A115</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fd</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> &amp; <font color="Olive" title="b4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">p2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c23">i</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> <font color="Maroon" title="c19">fd</font> implies  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</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="Olive" title="b2">p2</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_5_2"/><span class="lab"><font color="Green" title="E86">A116</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c23">i</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> <font color="Maroon" title="c19">fd</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</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="Olive" title="b2">p2</font> )</span><br/>

<a NAME="E2:35_1_5_2"/><span class="lab"><font color="Green" title="E87">A117</font></span>: 
<font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:35_1_5"><span class="lab"><font color="Green" title="E82">A110</font></span></a>, <a class="txt" href="#E1:35_1_5_2"><span class="lab"><font color="Green" title="E86">A116</font></span></a></span>;<br/>
<a NAME="E3:35_1_5_2"/><span class="lab"><font color="Green" title="E88">A118</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">i</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> <font color="Maroon" title="c19">fd</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_5_2"><span class="lab"><font color="Green" title="E86">A116</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E4:35_1_5_2"/><span class="lab"><font color="Green" title="E89">A119</font></span>: 
<font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">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> <font color="Maroon" title="c19">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:35_1_5"><span class="lab"><font color="Green" title="E82">A110</font></span></a>, <a class="txt" href="#E3:35_1_5_2"><span class="lab"><font color="Green" title="E88">A118</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:35_1_5_2"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</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="Olive" title="b2">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1_5"><span class="lab"><font color="Green" title="E72">A100</font></span></a>, <a class="txt" href="#E1:35_1_5_2"><span class="lab"><font color="Green" title="E86">A116</font></span></a>, <a class="txt" href="#E2:35_1_5_2"><span class="lab"><font color="Green" title="E87">A117</font></span></a>, <a class="txt" href="#E4:35_1_5_2"><span class="lab"><font color="Green" title="E89">A119</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E35:35_1_5"/><span class="lab"><font color="Green" title="E86">A120</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">f</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> &amp; <font color="Olive" title="b4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">p2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_5_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c23">i</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> <font color="Maroon" title="c21">f</font> implies  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</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="Olive" title="b2">p2</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_5_3"/><span class="lab"><font color="Green" title="E87">A121</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font>
 <span class="kw">and </span><br/><a NAME="E2:35_1_5_3"/><span class="lab"><font color="Green" title="E88">A122</font></span>: 
<font color="Maroon" title="c23">i</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> <font color="Maroon" title="c21">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</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="Olive" title="b2">p2</font> )</span><br/>

<a NAME="E3:35_1_5_3"/><span class="lab"><font color="Green" title="E89">A123</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fd</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1_5"><span class="lab"><font color="Green" title="E74">A102</font></span></a>, <a class="txt" href="#E2:35_1_5_3"><span class="lab"><font color="Green" title="E88">A122</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E90">A124</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_5_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_5_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_5_3_1_1"/>
( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c23">i</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> <font color="Maroon" title="c19">fd</font> ) or ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fd</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_5_3"><span class="lab"><font color="Green" title="E87">A121</font></span></a>, <a class="txt" href="#E3:35_1_5_3"><span class="lab"><font color="Green" title="E89">A123</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_5_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_5_3_1_1_1"/><span class="lab"><font color="Green" title="E91">A125</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c23">i</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> <font color="Maroon" title="c19">fd</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</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="Olive" title="b2">p2</font> )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:35_1_5_3_1_1_1"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</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="Olive" title="b2">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:35_1_5"><span class="lab"><font color="Green" title="E85">A115</font></span></a>, <a class="txt" href="#E1:35_1_5_3_1_1_1"><span class="lab"><font color="Green" title="E91">A125</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_5_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_5_3_1_1_2"/><span class="lab"><font color="Green" title="E91">A126</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fd</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</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="Olive" title="b2">p2</font> )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:35_1_5_3_1_1_2"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</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="Olive" title="b2">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1_5"><span class="lab"><font color="Green" title="E65">A93</font></span></a>, <a class="txt" href="#E13:35_1_5"><span class="lab"><font color="Green" title="E68">A96</font></span></a>, <a class="txt" href="#E25:35_1_5"><span class="lab"><font color="Green" title="E77">A105</font></span></a>, <a class="txt" href="#E29:35_1_5"><span class="lab"><font color="Green" title="E80">A108</font></span></a>, <a class="txt" href="#E33:35_1_5"><span class="lab"><font color="Green" title="E84">A114</font></span></a>, <a class="txt" href="#E1:35_1_5_3_1_1_2"><span class="lab"><font color="Green" title="E91">A126</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E5:35_1_5_3"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</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="Olive" title="b2">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_5_3"><span class="lab"><font color="Green" title="E90">A124</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E36:35_1_5"/><span class="lab"><font color="Green" title="E87">A127</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">f</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> &amp; <font color="Olive" title="b4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> &amp; <font color="Olive" title="b4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">p2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_5_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c23">i</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> <font color="Maroon" title="c21">f</font> implies  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</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="Olive" title="b2">p2</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_5_4"/><span class="lab"><font color="Green" title="E88">A128</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c23">i</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> <font color="Maroon" title="c21">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</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="Olive" title="b2">p2</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c24">p1</font>, <font color="Maroon" title="c25">p2</font>, <font color="Maroon" title="c26">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_5_4"/><span class="lab"><font color="Green" title="E89">A129</font></span>: 
( <font color="Maroon" title="c24">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Maroon" title="c25">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">p1</font> )
 <span class="kw">and </span><br/><a NAME="E4:35_1_5_4"/><span class="lab"><font color="Green" title="E90">A130</font></span>: 
<font color="Maroon" title="c26">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c25">p2</font>
 <span class="kw">and </span><br/><a NAME="E5:35_1_5_4"/><span class="lab"><font color="Green" title="E91">A131</font></span>: 
<font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</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="c25">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:35_1_5"><span class="lab"><font color="Green" title="E86">A120</font></span></a>, <a class="txt" href="#E1:35_1_5_4"><span class="lab"><font color="Green" title="E88">A128</font></span></a></span>;<br/>
<a NAME="E6:35_1_5_4"/><span class="lab"><font color="Green" title="E92">A132</font></span>: 
( <font color="Maroon" title="c26">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">p1</font> &amp; <font color="Maroon" title="c26">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">p2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_5_4"><span class="lab"><font color="Green" title="E90">A130</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">thus </span><a NAME="E7:35_1_5_4"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c21">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</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="Olive" title="b2">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_5_4"><span class="lab"><font color="Green" title="E89">A129</font></span></a>, <a class="txt" href="#E5:35_1_5_4"><span class="lab"><font color="Green" title="E91">A131</font></span></a>, <a class="txt" href="#E6:35_1_5_4"><span class="lab"><font color="Green" title="E92">A132</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E37:35_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E7:35_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E8:35_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E5:35_1_5"><span class="lab"><font color="Green" title="E62">A90</font></span></a>, <a class="txt" href="#E22:35_1_5"><span class="lab"><font color="Green" title="E74">A102</font></span></a>, <a class="txt" href="#E24:35_1_5"><span class="lab"><font color="Green" title="E76">A104</font></span></a>, <a class="txt" href="#E32:35_1_5"><span class="lab"><font color="Green" title="E83">A113</font></span></a>, <a class="txt" href="#E36:35_1_5"><span class="lab"><font color="Green" title="E87">A127</font></span></a>, <a class="ref" href="partit1.html#T25" target="_self" title="PARTIT1:th.25">Th25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E41:35_1"/><span class="lab"><font color="Green" title="E59">A133</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">pb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E6:35_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E34:35_1"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="txt" href="#E39:35_1"><span class="lab"><font color="Green" title="E57">A85</font></span></a>, <a class="txt" href="#E40:35_1"><span class="lab"><font color="Green" title="E58">A86</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E43:35_1"/><span class="lab"><font color="Green" title="E60">A134</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">y1</font>
 <span class="kw">and </span><br/><a NAME="E44:35_1"/><span class="lab"><font color="Green" title="E61">A135</font></span>: 
<font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Cb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:35_1"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="txt" href="#E41:35_1"><span class="lab"><font color="Green" title="E59">A133</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E45:35_1"/><span class="lab"><font color="Green" title="E62">A136</font></span>: 
 ex <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">PB</font> st <br/>( <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  ex <font color="Olive" title="b2">q</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ca</font> &amp; <font color="Olive" title="b1">p</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:35_1"><span class="lab"><font color="Green" title="E61">A135</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E47:35_1"/><span class="lab"><font color="Green" title="E63">A137</font></span>: 
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Ca</font>
 <span class="kw">and </span><br/><a NAME="E48:35_1"/><span class="lab"><font color="Green" title="E64">A138</font></span>: 
<font color="Maroon" title="c16">y1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:35_1"><span class="lab"><font color="Green" title="E62">A136</font></span></a></span>;<br/>
<a NAME="E49:35_1"/><span class="lab"><font color="Green" title="E65">A139</font></span>: 
 ex <font color="Olive" title="b1">pp</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">PA</font> st <br/>( <font color="Maroon" title="c17">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">pp</font> &amp;  ex <font color="Olive" title="b2">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Y</font> st <br/>( 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> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">pp</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font> holds <br/> ex <font color="Olive" title="b4">p1</font>, <font color="Olive" title="b5">p2</font>, <font color="Olive" title="b6">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b5">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">p1</font> &amp; <font color="Olive" title="b6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b5">p2</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">p2</font> ) ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:35_1"><span class="lab"><font color="Green" title="E63">A137</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">fd</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Y</font><span class="kw"> such that </span><br/><a NAME="E51:35_1"/><span class="lab"><font color="Green" title="E66">A140</font></span>: 
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> <font color="Maroon" title="c18">fd</font>
 <span class="kw">and </span><br/><a NAME="E52:35_1"/><span class="lab"><font color="Green" title="E67">A141</font></span>: 
<font color="Maroon" title="c18">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E53:35_1"/><span class="lab"><font color="Green" title="E68">A142</font></span>: 
<font color="Maroon" title="c18">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">fd</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">q</font>
 <span class="kw">and </span><br/><a NAME="E54:35_1"/><span class="lab"><font color="Green" title="E69">A143</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">fd</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c18">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> &amp; <font color="Olive" title="b4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Maroon" title="c18">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:35_1"><span class="lab"><font color="Green" title="E65">A139</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">y9</font> = <font color="Maroon" title="c5">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">f</font> = <font color="Maroon" title="c18">fd</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">y9</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">Y</font> ;<br/>
<a NAME="E57:35_1"/><span class="lab"><font color="Green" title="E70">A144</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">fd</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">y9</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/>
<a NAME="E58:35_1"/><span class="lab"><font color="Green" title="E71">A145</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">fd</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:35_1"><span class="lab"><font color="Green" title="E70">A144</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/>
<a NAME="E59:35_1"/><span class="lab"><font color="Green" title="E72">A146</font></span>: 
1 <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> <font color="Maroon" title="c20">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:35_1"><span class="lab"><font color="Green" title="E66">A140</font></span></a>, <a class="txt" href="#E58:35_1"><span class="lab"><font color="Green" title="E71">A145</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E60:35_1"/><span class="lab"><font color="Green" title="E73">A147</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">fd</font></span>)</span> <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="c16">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:35_1"><span class="lab"><font color="Green" title="E60">A134</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E61:35_1"/><span class="lab"><font color="Green" title="E74">A148</font></span>: 
<font color="Maroon" title="c16">y1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:35_1"><span class="lab"><font color="Green" title="E64">A138</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">z0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E63:35_1"/><span class="lab"><font color="Green" title="E75">A149</font></span>: 
( <font color="Maroon" title="c21">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">y1</font> &amp; <font color="Maroon" title="c21">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E61:35_1"><span class="lab"><font color="Green" title="E74">A148</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E64:35_1"/><span class="lab"><font color="Green" title="E76">A150</font></span>: 
<font color="Maroon" title="c21">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">y1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:35_1"><span class="lab"><font color="Green" title="E75">A149</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E65:35_1"/><span class="lab"><font color="Green" title="E77">A151</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">fd</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">fd</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E66:35_1"/><span class="lab"><font color="Green" title="E78">A152</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">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> <font color="Maroon" title="c18">fd</font> holds <br/><font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">k</font> &amp; <font color="Maroon" title="c22">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> <font color="Maroon" title="c18">fd</font> implies <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_6"/><span class="lab"><font color="Green" title="E79">A153</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">k</font> &amp; <font color="Maroon" title="c22">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> <font color="Maroon" title="c18">fd</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font></span><br/>

<a NAME="E2:35_1_6"/><span class="lab"><font color="Green" title="E80">A154</font></span>: 
<font color="Maroon" title="c22">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> <font color="Maroon" title="c18">fd</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:35_1"><span class="lab"><font color="Green" title="E77">A151</font></span></a>, <a class="txt" href="#E1:35_1_6"><span class="lab"><font color="Green" title="E79">A153</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:35_1_6"/>
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_6"><span class="lab"><font color="Green" title="E80">A154</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E67:35_1"/><span class="lab"><font color="Green" title="E79">A155</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">fd</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:35_1"><span class="lab"><font color="Green" title="E66">A140</font></span></a>, <a class="txt" href="#E53:35_1"><span class="lab"><font color="Green" title="E68">A142</font></span></a>, <a class="txt" href="#E66:35_1"><span class="lab"><font color="Green" title="E78">A152</font></span></a></span>;<br/>
<a NAME="E68:35_1"/><span class="lab"><font color="Green" title="E80">A156</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">fd</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> &amp; <font color="Olive" title="b4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">p2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c22">i</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> <font color="Maroon" title="c18">fd</font> implies  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</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="Olive" title="b2">p2</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_7"/><span class="lab"><font color="Green" title="E81">A157</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c22">i</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> <font color="Maroon" title="c18">fd</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</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="Olive" title="b2">p2</font> )</span><br/>

<a NAME="E2:35_1_7"/><span class="lab"><font color="Green" title="E82">A158</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:35_1"><span class="lab"><font color="Green" title="E78">A152</font></span></a>, <a class="txt" href="#E1:35_1_7"><span class="lab"><font color="Green" title="E81">A157</font></span></a></span>;<br/>
<a NAME="E3:35_1_7"/><span class="lab"><font color="Green" title="E83">A159</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">i</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> <font color="Maroon" title="c18">fd</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_7"><span class="lab"><font color="Green" title="E81">A157</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E4:35_1_7"/><span class="lab"><font color="Green" title="E84">A160</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">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> <font color="Maroon" title="c18">fd</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:35_1"><span class="lab"><font color="Green" title="E78">A152</font></span></a>, <a class="txt" href="#E3:35_1_7"><span class="lab"><font color="Green" title="E83">A159</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:35_1_7"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</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="Olive" title="b2">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:35_1"><span class="lab"><font color="Green" title="E69">A143</font></span></a>, <a class="txt" href="#E1:35_1_7"><span class="lab"><font color="Green" title="E81">A157</font></span></a>, <a class="txt" href="#E2:35_1_7"><span class="lab"><font color="Green" title="E82">A158</font></span></a>, <a class="txt" href="#E4:35_1_7"><span class="lab"><font color="Green" title="E84">A160</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E69:35_1"/><span class="lab"><font color="Green" title="E81">A161</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">f</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font>, <font color="Olive" title="b4">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> &amp; <font color="Olive" title="b4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">p2</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">p2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c22">i</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> <font color="Maroon" title="c20">f</font> implies  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</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="Olive" title="b2">p2</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1_8"/><span class="lab"><font color="Green" title="E82">A162</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">and </span><br/><a NAME="E2:35_1_8"/><span class="lab"><font color="Green" title="E83">A163</font></span>: 
<font color="Maroon" title="c22">i</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> <font color="Maroon" title="c20">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</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="Olive" title="b2">p2</font> )</span><br/>

<a NAME="E3:35_1_8"/><span class="lab"><font color="Green" title="E84">A164</font></span>: 
<font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">fd</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1"><span class="lab"><font color="Green" title="E71">A145</font></span></a>, <a class="txt" href="#E2:35_1_8"><span class="lab"><font color="Green" title="E83">A163</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E85">A165</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_8_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_8_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_8_1_1"/>
( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c22">i</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> <font color="Maroon" title="c18">fd</font> ) or ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">fd</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_8"><span class="lab"><font color="Green" title="E82">A162</font></span></a>, <a class="txt" href="#E3:35_1_8"><span class="lab"><font color="Green" title="E84">A164</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_8_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_8_1_1_1"/><span class="lab"><font color="Green" title="E86">A166</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c22">i</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> <font color="Maroon" title="c18">fd</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</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="Olive" title="b2">p2</font> )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:35_1_8_1_1_1"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</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="Olive" title="b2">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:35_1"><span class="lab"><font color="Green" title="E80">A156</font></span></a>, <a class="txt" href="#E1:35_1_8_1_1_1"><span class="lab"><font color="Green" title="E86">A166</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_8_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_8_1_1_2"/><span class="lab"><font color="Green" title="E86">A167</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">fd</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</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="Olive" title="b2">p2</font> )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:35_1_8_1_1_2"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</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="Olive" title="b2">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:35_1"><span class="lab"><font color="Green" title="E62">A136</font></span></a>, <a class="txt" href="#E49:35_1"><span class="lab"><font color="Green" title="E65">A139</font></span></a>, <a class="txt" href="#E60:35_1"><span class="lab"><font color="Green" title="E73">A147</font></span></a>, <a class="txt" href="#E64:35_1"><span class="lab"><font color="Green" title="E76">A150</font></span></a>, <a class="txt" href="#E67:35_1"><span class="lab"><font color="Green" title="E79">A155</font></span></a>, <a class="txt" href="#E1:35_1_8_1_1_2"><span class="lab"><font color="Green" title="E86">A167</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E5:35_1_8"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</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="Olive" title="b2">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_8"><span class="lab"><font color="Green" title="E85">A165</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E70:35_1"/><span class="lab"><font color="Green" title="E82">A168</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c18">fd</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c19">y9</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">fd</font></span>)</span> <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="c5">y</font> &amp; 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> <font color="Maroon" title="c20">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:35_1"><span class="lab"><font color="Green" title="E72">A146</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E71:35_1"/><span class="lab"><font color="Green" title="E83">A169</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:35_1"><span class="lab"><font color="Green" title="E66">A140</font></span></a>, <a class="txt" href="#E52:35_1"><span class="lab"><font color="Green" title="E67">A141</font></span></a>, <a class="txt" href="#E66:35_1"><span class="lab"><font color="Green" title="E78">A152</font></span></a></span>;<br/>
<a NAME="E72:35_1"/><span class="lab"><font color="Green" title="E84">A170</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">f</font> holds <br/> ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</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="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Olive" title="b2">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">u</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<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></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> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c22">i</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> <font color="Maroon" title="c20">f</font> implies  ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</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="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span>,<font color="Olive" title="b1">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">u</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">i</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> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_9"/><span class="lab"><font color="Green" title="E85">A171</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c22">i</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> <font color="Maroon" title="c20">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</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="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span>,<font color="Olive" title="b1">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">u</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">i</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> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c23">p1</font>, <font color="Maroon" title="c24">p2</font>, <font color="Maroon" title="c25">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:35_1_9"/><span class="lab"><font color="Green" title="E86">A172</font></span>: 
<font color="Maroon" title="c23">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font>
 <span class="kw">and </span><br/><a NAME="E4:35_1_9"/><span class="lab"><font color="Green" title="E87">A173</font></span>: 
<font color="Maroon" title="c24">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font>
 <span class="kw">and </span><br/><a NAME="E5:35_1_9"/><span class="lab"><font color="Green" title="E88">A174</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">p1</font>
 <span class="kw">and </span><br/><a NAME="E6:35_1_9"/><span class="lab"><font color="Green" title="E89">A175</font></span>: 
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">p1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c24">p2</font>
 <span class="kw">and </span><br/><a NAME="E7:35_1_9"/><span class="lab"><font color="Green" title="E90">A176</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</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="c24">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:35_1"><span class="lab"><font color="Green" title="E81">A161</font></span></a>, <a class="txt" href="#E1:35_1_9"><span class="lab"><font color="Green" title="E85">A171</font></span></a></span>;<br/>
<a NAME="E8:35_1_9"/><span class="lab"><font color="Green" title="E91">A177</font></span>: 
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_9"><span class="lab"><font color="Green" title="E89">A175</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E9:35_1_9"/><span class="lab"><font color="Green" title="E92">A178</font></span>: 
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_1_9"><span class="lab"><font color="Green" title="E89">A175</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">x2</font> = <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">y2</font> = <font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E12:35_1_9"/><span class="lab"><font color="Green" title="E93">A179</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">x2</font>,<font color="Maroon" title="c25">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_9"><span class="lab"><font color="Green" title="E86">A172</font></span></a>, <a class="txt" href="#E5:35_1_9"><span class="lab"><font color="Green" title="E88">A174</font></span></a>, <a class="txt" href="#E8:35_1_9"><span class="lab"><font color="Green" title="E91">A177</font></span></a>, <a class="ref" href="partit1.html#D6" target="_self" title="PARTIT1:def.6">Def6</a></span>;<br/>
<a NAME="E13:35_1_9"/><span class="lab"><font color="Green" title="E94">A180</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">u</font>,<font color="Maroon" title="c27">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_9"><span class="lab"><font color="Green" title="E87">A173</font></span></a>, <a class="txt" href="#E7:35_1_9"><span class="lab"><font color="Green" title="E90">A176</font></span></a>, <a class="txt" href="#E9:35_1_9"><span class="lab"><font color="Green" title="E92">A178</font></span></a>, <a class="ref" href="partit1.html#D6" target="_self" title="PARTIT1:def.6">Def6</a></span>;<br/>
<a NAME="E14:35_1_9"/><span class="lab"><font color="Green" title="E95">A181</font></span>: 
(  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> &amp;  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<span class="kw">thus </span><a NAME="E15:35_1_9"/>
 ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</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="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span>,<font color="Olive" title="b1">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">u</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">i</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> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1_9"><span class="lab"><font color="Green" title="E86">A172</font></span></a>, <a class="txt" href="#E8:35_1_9"><span class="lab"><font color="Green" title="E91">A177</font></span></a>, <a class="txt" href="#E12:35_1_9"><span class="lab"><font color="Green" title="E93">A179</font></span></a>, <a class="txt" href="#E13:35_1_9"><span class="lab"><font color="Green" title="E94">A180</font></span></a>, <a class="txt" href="#E14:35_1_9"><span class="lab"><font color="Green" title="E95">A181</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E73:35_1"/><span class="lab"><font color="Green" title="E85">A182</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x9</font>,<font color="Maroon" title="c19">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:35_1"><span class="lab"><font color="Green" title="E71">A145</font></span></a>, <a class="txt" href="#E70:35_1"><span class="lab"><font color="Green" title="E82">A168</font></span></a>, <a class="txt" href="#E71:35_1"><span class="lab"><font color="Green" title="E83">A169</font></span></a>, <a class="txt" href="#E72:35_1"><span class="lab"><font color="Green" title="E84">A170</font></span></a>, <a class="ref" href="partit1.html#T26" target="_self" title="PARTIT1:th.26">Th26</a></span>;<br/>
<span class="kw">thus </span><a NAME="E74:35_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E73:35_1"><span class="lab"><font color="Green" title="E85">A182</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:35"/><span class="lab"><font color="Green" title="E30">A183</font></span>: 
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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_2"/><span class="lab"><font color="Green" title="E31">A184</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span></span><br/>

<a NAME="E2:35_2"/><span class="lab"><font color="Green" title="E32">A185</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2"><span class="lab"><font color="Green" title="E31">A184</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:35_2"/><span class="lab"><font color="Green" title="E33">A186</font></span>: 
(  ex <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) or  ex <font color="Olive" title="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_2"><span class="lab"><font color="Green" title="E32">A185</font></span></a>, <a class="ref" href="partit1.html#D6" target="_self" title="PARTIT1:def.6">Def6</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E34">A187</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_2_1_1"/>
( ( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> &amp;  ex <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) ) or ( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> &amp;  ex <font color="Olive" title="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> ) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_2"><span class="lab"><font color="Green" title="E33">A186</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:35_2_1_1_1"/><span class="lab"><font color="Green" title="E35">A188</font></span>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> &amp;  ex <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c6">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><span class="kw"> such that </span><br/><a NAME="E3:35_2_1_1_1"/><span class="lab"><font color="Green" title="E36">A189</font></span>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font>
 <span class="kw">and </span><br/><a NAME="E4:35_2_1_1_1"/><span class="lab"><font color="Green" title="E37">A190</font></span>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2_1_1_1"><span class="lab"><font color="Green" title="E35">A188</font></span></a></span>;<br/><a NAME="E5:35_2_1_1_1"/><span class="lab"><font color="Green" title="E38">A191</font></span>: 
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E3:35_2_1_1_1"><span class="lab"><font color="Green" title="E36">A189</font></span></a>, <a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></span>;<br/><span class="kw">thus </span><a NAME="E6:35_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_2_1_1_1"><span class="lab"><font color="Green" title="E37">A190</font></span></a>, <a class="txt" href="#E5:35_2_1_1_1"><span class="lab"><font color="Green" title="E38">A191</font></span></a>, <a class="ref" href="partit1.html#D6" target="_self" title="PARTIT1:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:35_2_1_1_2"/><span class="lab"><font color="Green" title="E35">A192</font></span>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> &amp;  ex <font color="Olive" title="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c6">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><span class="kw"> such that </span><br/><a NAME="E3:35_2_1_1_2"/><span class="lab"><font color="Green" title="E36">A193</font></span>: 
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">PB</font>
 <span class="kw">and </span><br/><a NAME="E4:35_2_1_1_2"/><span class="lab"><font color="Green" title="E37">A194</font></span>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2_1_1_2"><span class="lab"><font color="Green" title="E35">A192</font></span></a></span>;<br/><a NAME="E5:35_2_1_1_2"/><span class="lab"><font color="Green" title="E38">A195</font></span>: 
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font> &amp; <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E3:35_2_1_1_2"><span class="lab"><font color="Green" title="E36">A193</font></span></a>, <a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></span>;<br/><span class="kw">thus </span><a NAME="E6:35_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_2_1_1_2"><span class="lab"><font color="Green" title="E37">A194</font></span></a>, <a class="txt" href="#E5:35_2_1_1_2"><span class="lab"><font color="Green" title="E38">A195</font></span></a>, <a class="ref" href="partit1.html#D6" target="_self" title="PARTIT1:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E5:35_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_2"><span class="lab"><font color="Green" title="E34">A187</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:35"/><span class="lab"><font color="Green" title="E31">A196</font></span>: 
<span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E30">A183</font></span></a>, <a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></span>;<br/>
<a NAME="E8:35"/><span class="lab"><font color="Green" title="E32">A197</font></span>: 
<span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35"><span class="lab"><font color="Green" title="E31">A196</font></span></a>, <a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:35"/>
 <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">PA</font> <a href="partit1.html#K3" title="PARTIT1:func.3">'\/'</a> <font color="Maroon" title="c3">PB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">PA</font></span>)</span> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">PB</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E8:35"><span class="lab"><font color="Green" title="E32">A197</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
