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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <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>, <font color="Olive" title="b2">q</font> being  <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> holds <br/> ( <span class="p1"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> iff ( <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> &amp; <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Olive" title="b2">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> iff ( <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> &amp; <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> ) )</span><br/>



<span class="kw">thus </span><a NAME="E1:63"/>
( <span class="p1"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> implies ( <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> &amp; <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</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"> ( <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> &amp; <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> implies <span class="p1"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:63_1"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E2:63_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A2</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:63_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<span class="p1"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</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"> ( <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> &amp; <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> )</span><br/>

<a NAME="E4:63_1"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<span class="p1"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></span>;<br/>
<a NAME="E5:63_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</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="#E3:63_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E2:63_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<a NAME="E6:63_1"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c2">p</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>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</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="#E3:63_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E2:63_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:63_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<a NAME="E7:63_1"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E2:63_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:63_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<a NAME="E8:63_1"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E2:63_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:63_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:63_1"/>
<span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</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"> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</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="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> or (  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</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> &amp;  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</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="E1:63_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 ; <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#K6" title="PSCOMP_1:func.6">W-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</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> &amp;  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</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/>

<a NAME="E2:63_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default">1</span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></span>;<br/>
<a NAME="E3:63_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:63_1_1"/>
(  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</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> &amp;  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</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="#E8:63_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E5:63_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E7:63_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E6:63_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</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">let </span><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</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="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> or (  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</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> &amp;  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</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="E10:63_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 ; <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#K6" title="PSCOMP_1:func.6">W-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</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> &amp;  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</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/>

<a NAME="E11:63_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default">1</span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></span>;<br/>
<a NAME="E12:63_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E13:63_1"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<span class="p1"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></span>;<br/>
<a NAME="E14:63_1"/><span class="lab"><font color="Green" title="E33">A11</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E15:63_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A12</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</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="#E3:63_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E13:63_1"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<a NAME="E16:63_1"/><span class="lab"><font color="Green" title="E35">A13</font></span>: 
<font color="Maroon" title="c3">q</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>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</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="#E3:63_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E14:63_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E13:63_1"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<a NAME="E17:63_1"/><span class="lab"><font color="Green" title="E36">A14</font></span>: 
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E14:63_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E13:63_1"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<a NAME="E18:63_1"/>
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E14:63_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E13:63_1"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:63_1"/>
(  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</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> &amp;  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</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="#E15:63_1"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E17:63_1"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E16:63_1"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E12:63_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</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="E2:63"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></span>;<br/>
<a NAME="E3:63"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default">1</span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></span>;<br/>
<a NAME="E4:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A16</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:63"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default">1</span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></span>;<br/>
<a NAME="E6:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A17</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:63"/><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</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"> (  not <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> or <span class="p1"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> )</span><br/>

<a NAME="E8:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A19</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</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="#E6:63"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<a NAME="E9:63"/><span class="lab"><font color="Green" title="E28">A20</font></span>: 
<font color="Maroon" title="c2">p</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>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</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="#E7:63"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E6:63"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<a NAME="E10:63"/><span class="lab"><font color="Green" title="E29">A21</font></span>: 
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E6:63"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<a NAME="E11:63"/><span class="lab"><font color="Green" title="E30">A22</font></span>: 
<span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></span>;<br/>
<span class="kw">assume </span><a NAME="E12:63"/><span class="lab"><font color="Green" title="E31">A23</font></span>: 
<span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</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"> <span class="p1"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font></span><br/>

<a NAME="E13:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A24</font></span>: 
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E11:63"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<a NAME="E14:63"/><span class="lab"><font color="Green" title="E33">A25</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</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="#E12:63"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E11:63"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</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="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> or (  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</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> &amp;  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</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="E15:63"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span>
 ; <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#K6" title="PSCOMP_1:func.6">W-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</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> &amp;  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</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/>

<a NAME="E16:63"/><span class="kw">then </span>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E17:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A26</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E18:63"/><span class="lab"><font color="Green" title="E35">A27</font></span>: 
<font color="Maroon" title="c3">q</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>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</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="#E12:63"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E11:63"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<a NAME="E19:63"/><span class="lab"><font color="Green" title="E36">A28</font></span>: 
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:63"><span class="lab"><font color="Green" title="E31">A23</font></span></a>, <a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E11:63"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<a NAME="E20:63"/>
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E6:63"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:63"/>
(  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</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> &amp;  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</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="#E8:63"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E10:63"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E9:63"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E13:63"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="txt" href="#E14:63"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E19:63"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="txt" href="#E18:63"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E17:63"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</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>
