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

<span class="kw">let </span><font color="Maroon" title="c1">f</font> be  non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>   <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">g</font> =  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:40"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="revrot_1.html#T33" title="REVROT_1:th.33">REVROT_1:33</a></span>;<br/>
<a NAME="E2:40"/>
 <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_2.html#T43" title="SPRECT_2:th.43">SPRECT_2:43</a></span>;<br/>
<a NAME="E3:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c1">f</font>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="finseq_6.html#T98" title="FINSEQ_6:th.98">FINSEQ_6:98</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:40"/>
<a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:40_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T31" target="_self" title="GOBRD14:th.31">Th31</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">a</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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> or <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:40_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 ; <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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:40_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:40_1"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:40_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_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="E1:40_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_1"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">b</font> = <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>;<br/><a NAME="E3:40_1_1_1"/>
( <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:40_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="jordan2c.html#T119" title="JORDAN2C:th.119">JORDAN2C:119</a></span>;<br/><a NAME="E5:40_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="revrot_1.html#T36" title="REVROT_1:th.36">REVROT_1:36</a></span>;<br/><a NAME="E6:40_1_1_1"/><span class="kw">then </span>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c5">x</font>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="gobrd14.html#T37" target="_self" title="GOBRD14:th.37">Th37</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:40_1_1_1"/><span class="lab"><font color="Green" title="E28">A8</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c5">x</font>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E9:40_1_1_1"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c7">c</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40_1_1_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a></span>;<br/><a NAME="E11:40_1_1_1"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>;<br/><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">c</font> = 
<font color="Maroon" title="c8">c</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></span>
<br/>.= 
<font color="Maroon" title="c5">x</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40_1_1_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="#E11:40_1_1_1"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="ref" href="goboard7.html#T6" title="GOBOARD7:th.6">GOBOARD7:6</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></span>
;<br/><span class="kw">hence </span><a NAME="E13:40_1_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40_1_1_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</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="40_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</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>

<a NAME="E5:40"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T29" target="_self" title="GOBRD14:th.29">Th29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:40"/>
<a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</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>
