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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <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="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> implies (  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp;  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">K</font> =  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">and </span><br/><a NAME="E2:71"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font>
 ; <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#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp;  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )</span><br/>

<a NAME="E3:71"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
 <a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c6">X</font> =  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">Z</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E2:71_1"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></span>;<br/>
<a NAME="E3:71_1"/><span class="lab"><font color="Green" title="E63">A5</font></span>: 
for <font color="Olive" title="b1">p</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">p</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <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="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> implies <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">c</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:71_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</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="c8">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">c</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">p0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:71_1_1"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
<font color="Maroon" title="c9">p0</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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:71_1_1"/>
<font color="Maroon" title="c9">p0</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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:71_1_1"/><span class="lab"><font color="Green" title="E65">A7</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p0</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">p0</font> = <font color="Maroon" title="c9">p0</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="#E2:71_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="#E3:71_1_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a></span>;<br/>
<a NAME="E7:71_1_1"/><span class="lab"><font color="Green" title="E66">A8</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></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="c3">c</font>
 
<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 NAME="E8:71_1_1"/>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></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="c4">d</font>
 
<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 NAME="E9:71_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">p0</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E2:71_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="#E3:71_1_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E7:71_1_1"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:71_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="#E3:71_1_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="txt" href="#E5:71_1_1"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="ref" href="pscomp_1.html#T70" title="PSCOMP_1:th.70">PSCOMP_1:70</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="E4:71_1"/><span class="lab"><font color="Green" title="E64">A9</font></span>: 
for <font color="Olive" title="b1">q</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st ( for <font color="Olive" title="b2">p</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">q</font> ) holds <br/><font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">p</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">q</font> ) implies <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:71_1_2"/><span class="lab"><font color="Green" title="E65">A10</font></span>: 
for <font color="Olive" title="b1">p</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">q</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">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">q</font></span><br/>

<a NAME="E2:71_1_2"/><span class="lab"><font color="Green" title="E66">A11</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></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="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T69" title="RLTOPSP1:th.69">RLTOPSP1:69</a></span>;<br/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> = 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <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#T70" title="PSCOMP_1:th.70">PSCOMP_1:70</a>, <a class="ref" href="rltopsp1.html#T69" title="RLTOPSP1:th.69">RLTOPSP1:69</a></span>
<br/>.= 
<font color="Maroon" title="c3">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:71_1_2"/>
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="txt" href="#E1:71_1_2"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="txt" href="#E2:71_1_2"><span class="lab"><font color="Green" title="E66">A11</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>
<span class="kw">thus </span> <a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span> = 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c7">Z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#D20" title="PSCOMP_1:def.20">PSCOMP_1:def 20</a></span>
<br/>.= 
<font color="Maroon" title="c3">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E4:71_1"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="ref" href="seq_4.html#T61" title="SEQ_4:th.61">SEQ_4:61</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="E4:71"/><span class="lab"><font color="Green" title="E62">A12</font></span>: 
 <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">W-most</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="jgraph_6.html#T54" target="_self" title="JGRAPH_6:th.54">Th54</a></span>;<br/>
<a NAME="E5:71"/><span class="lab"><font color="Green" title="E63">A13</font></span>: 
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="jgraph_6.html#T46" target="_self" title="JGRAPH_6:th.46">Th46</a></span>;<br/>
<a NAME="E6:71"/><span class="lab"><font color="Green" title="E64">A14</font></span>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c6">X</font> =  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">Z</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E2:71_2"/><span class="lab"><font color="Green" title="E65">A15</font></span>: 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></span>;<br/>
<a NAME="E3:71_2"/><span class="lab"><font color="Green" title="E66">A16</font></span>: 
for <font color="Olive" title="b1">p</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">p</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <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="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> implies <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:71_2_1"/>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</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="c8">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">p0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:71_2_1"/><span class="lab"><font color="Green" title="E67">A17</font></span>: 
<font color="Maroon" title="c9">p0</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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:71_2_1"/>
<font color="Maroon" title="c9">p0</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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:71_2_1"/><span class="lab"><font color="Green" title="E68">A18</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p0</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">p0</font> = <font color="Maroon" title="c9">p0</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="#E2:71_2"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="txt" href="#E3:71_2_1"><span class="lab"><font color="Green" title="E67">A17</font></span></a></span>;<br/>
<a NAME="E7:71_2_1"/><span class="lab"><font color="Green" title="E69">A19</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></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="c3">c</font>
 
<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 NAME="E8:71_2_1"/>
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></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="c4">d</font>
 
<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 NAME="E9:71_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">p0</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E2:71_2"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="txt" href="#E3:71_2_1"><span class="lab"><font color="Green" title="E67">A17</font></span></a>, <a class="txt" href="#E7:71_2_1"><span class="lab"><font color="Green" title="E69">A19</font></span></a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:71_2_1"/>
<font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_2"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="txt" href="#E3:71_2_1"><span class="lab"><font color="Green" title="E67">A17</font></span></a>, <a class="txt" href="#E5:71_2_1"><span class="lab"><font color="Green" title="E68">A18</font></span></a>, <a class="ref" href="pscomp_1.html#T70" title="PSCOMP_1:th.70">PSCOMP_1:70</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="E4:71_2"/><span class="lab"><font color="Green" title="E67">A20</font></span>: 
for <font color="Olive" title="b1">q</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st ( for <font color="Olive" title="b2">p</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> ) holds <br/><font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">p</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">q</font> ) implies <font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:71_2_2"/><span class="lab"><font color="Green" title="E68">A21</font></span>: 
for <font color="Olive" title="b1">p</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Z</font> holds <br/><font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">q</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="c4">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">q</font></span><br/>

<a NAME="E2:71_2_2"/><span class="lab"><font color="Green" title="E69">A22</font></span>: 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></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="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T69" title="RLTOPSP1:th.69">RLTOPSP1:69</a></span>;<br/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> = 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <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#T70" title="PSCOMP_1:th.70">PSCOMP_1:70</a>, <a class="ref" href="rltopsp1.html#T69" title="RLTOPSP1:th.69">RLTOPSP1:69</a></span>
<br/>.= 
<font color="Maroon" title="c4">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:71_2_2"/>
<font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_2"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="txt" href="#E1:71_2_2"><span class="lab"><font color="Green" title="E68">A21</font></span></a>, <a class="txt" href="#E2:71_2_2"><span class="lab"><font color="Green" title="E69">A22</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>
<span class="kw">thus </span> <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">proj2</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span>)</span></span>)</span> = 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c7">Z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#D21" title="PSCOMP_1:def.21">PSCOMP_1:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c4">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_2"><span class="lab"><font color="Green" title="E66">A16</font></span></a>, <a class="txt" href="#E4:71_2"><span class="lab"><font color="Green" title="E67">A20</font></span></a>, <a class="ref" href="seq_4.html#T63" title="SEQ_4:th.63">SEQ_4:63</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="E7:71"/><span class="lab"><font color="Green" title="E65">A23</font></span>: 
 <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">E-most</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="jgraph_6.html#T55" target="_self" title="JGRAPH_6:th.55">Th55</a></span>;<br/>
<a NAME="E8:71"/>
 <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="ref" href="jgraph_6.html#T48" target="_self" title="JGRAPH_6:th.48">Th48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:71"/>
(  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp;  <a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="txt" href="#E6:71"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="txt" href="#E7:71"><span class="lab"><font color="Green" title="E65">A23</font></span></a>, <a class="ref" href="pscomp_1.html#D42" title="PSCOMP_1:def.42">PSCOMP_1:def 42</a>, <a class="ref" href="pscomp_1.html#D46" title="PSCOMP_1:def.46">PSCOMP_1:def 46</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>
