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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp;  ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</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 <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2 );<br/>
<div><i><font color="Green" title="E10">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>] )</span><br/><b>assume </b><a NAME="E1:31_1_1_1"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>]</span><br/><a NAME="E2:31_1_1_1"/>
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 ;<br/><b>hence </b><a NAME="E3:31_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>]
 <b>by </b><i><a class="txt" href="#E1:31_1_1_1"><i><font color="Green" title="E11">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E2:31_1_1"/><i><font color="Green" title="E11">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>]
 
;<br/>
<div><i><font color="Green" title="E12">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font>] )</span><br/><b>assume </b><a NAME="E1:31_1_1_2"/><i><font color="Green" title="E13">A5</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font>]</span><br/><b>then consider </b><font color="Maroon" title="c6">k</font>, <font color="Maroon" title="c7">l</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> <b> such that </b><br/><a NAME="E3:31_1_1_2"/><i><font color="Green" title="E14">A6</font></i>: 
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font>
 ;<br/><b>consider </b><font color="Maroon" title="c8">m</font>, <font color="Maroon" title="c9">n</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> <b> such that </b><br/><a NAME="E5:31_1_1_2"/><i><font color="Green" title="E15">A7</font></i>: 
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="#E1:31_1_1_2"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/><a NAME="E6:31_1_1_2"/><i><font color="Green" title="E16">A8</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></i>;<br/><a NAME="E7:31_1_1_2"/><i><font color="Green" title="E17">A9</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></i>;<br/><a NAME="E8:31_1_1_2"/><i><font color="Green" title="E18">A10</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></i>;<br/><a NAME="E9:31_1_1_2"/><i><font color="Green" title="E19">A11</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></i>;<br/><span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">m</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79" title="FUNCT_7:th.79">FUNCT_7:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">m</font></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="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:31_1_1_2"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E3:31_1_1_2"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="#E6:31_1_1_2"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">m</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">l</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font>
<b>by </b><i><a class="txt" href="#E1:31_1_1_2"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E7:31_1_1_2"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">l</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79" title="FUNCT_7:th.79">FUNCT_7:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79" title="FUNCT_7:th.79">FUNCT_7:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">l</font></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="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:31_1_1_2"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E5:31_1_1_2"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="#E8:31_1_1_2"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font>
<b>by </b><i><a class="txt" href="#E1:31_1_1_2"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="#E9:31_1_1_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font>
<b>by </b><i><a class="ref" href="funct_7.html#T79" title="FUNCT_7:th.79">FUNCT_7:79</a></i>
;<br/><b>hence </b><a NAME="E11:31_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font>]
 <b>by </b><i><a class="txt" href="#E1:31_1_1_2"><i><font color="Green" title="E13">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">EqR</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E5:31_1_1"/><i><font color="Green" title="E13">A12</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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> <font color="Maroon" title="c3">EqR</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>from </b><i><a class="ref" href="eqrel_1.html#S1" title="EQREL_1:sch.1">EQREL_1:sch 1</a>(<a class="txt" href="#E1:31_1_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E2:31_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E3:31_1_1"><i><font color="Green" title="E12">A4</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c3">EqR</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> holds <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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> <font color="Maroon" title="c3">EqR</font> iff  ex <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">l</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 <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b4">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> )</span><br/>

<b>let </b><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> implies ( <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> <font color="Maroon" title="c3">EqR</font> iff  ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</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 <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> ) )</span><br/>



<b>assume </b><a NAME="E6:31_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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> <font color="Maroon" title="c3">EqR</font> iff  ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</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 <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )</span><br/>

<b>hence </b><a NAME="E7:31_1_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> <font color="Maroon" title="c3">EqR</font> iff  ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</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 <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abian.html#K1" title="ABIAN:func.1">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="txt" href="#E5:31_1_1"><i><font color="Green" title="E13">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
