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<b>let </b><font color="Maroon" title="c1">f</font> be  <a href="sprect_1.html#V1" title="SPRECT_1:attr.1">rectangular</a> <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>;<br/><b>let </b><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>;<br/>



<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E28">A1</font></i>: 
not <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font>
 ;<br/>

<b>assume </b><a NAME="E2:53"/><i><font color="Green" title="E29">A2</font></i>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&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>
 ;<br/>

<a NAME="E3:53"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="sprect_3.html#T59" target="_self" title="SPRECT_3:th.59">Th59</a></i>;<br/>
<a NAME="E4:53"/><b>then </b><i><font color="Green" title="E30">A3</font></i>: 
( <font color="Maroon" title="c3">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c3">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c3">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c3">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:53"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="sprect_3.html#T60" target="_self" title="SPRECT_3:th.60">Th60</a></i>;<br/>
<a NAME="E5:53"/><i><font color="Green" title="E31">A4</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&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>
 
<b>by </b><i><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></i>;<br/>
<a NAME="E6:53"/><b>then </b><i><font color="Green" title="E32">A5</font></i>: 
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="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E7:53"/><i><font color="Green" title="E33">A6</font></i>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/>
<a NAME="E8:53"/><b>then </b><i><font color="Green" title="E34">A7</font></i>: 
(  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">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#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:53"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="sprect_3.html#E6:53"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></i>;<br/>
<a NAME="E9:53"/><i><font color="Green" title="E35">A8</font></i>: 
(  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">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#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:53"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="sprect_3.html#E6:53"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="sprect_3.html#E7:53"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></i>;<br/>
<a NAME="E10:53"/><i><font color="Green" title="E36">A9</font></i>: 
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="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E5:53"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E11:53"/><i><font color="Green" title="E37">A10</font></i>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/>
<a NAME="E12:53"/><b>then </b>
(  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">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#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c3">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:53"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="sprect_3.html#E10:53"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></i>;<br/>
<a NAME="E13:53"/><b>then </b><i><font color="Green" title="E38">A11</font></i>: 
(  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c3">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E4:53"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E14:53"/>
(  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">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#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c3">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:53"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="sprect_3.html#E10:53"><i><font color="Green" title="E36">A9</font></i></a>, <a class="txt" href="sprect_3.html#E11:53"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></i>;<br/>
<a NAME="E15:53"/><b>then </b><i><font color="Green" title="E39">A12</font></i>: 
(  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c3">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E4:53"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E16:53"/><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 ;<br/>

<a NAME="E17:53"/><b>then </b><i><font color="Green" title="E41">A14</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">p'</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="sprect_3.html#E16:53"><i><font color="Green" title="E40">A13</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E20:53"/><i><font color="Green" title="E42">A15</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font>
 <b>and </b><br/><a NAME="E21:53"/><i><font color="Green" title="E43">A16</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>and </b><br/><a NAME="E22:53"/><i><font color="Green" title="E44">A17</font></i>: 
<font color="Maroon" title="c5">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E17:53"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="sppol_1.html#T21" title="SPPOL_1:th.21">SPPOL_1:21</a></i>;<br/>
<i><font color="Green" title="E45">A18</font></i>: <font color="Maroon" title="c5">p'</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E22:53"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="topreal3.html#T7" title="TOPREAL3:th.7">TOPREAL3:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9" title="TOPREAL3:th.9">TOPREAL3:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9" title="TOPREAL3:th.9">TOPREAL3:9</a></i>
;<br/>
<i><font color="Green" title="E46">A19</font></i>: <font color="Maroon" title="c5">p'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E22:53"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="topreal3.html#T7" title="TOPREAL3:th.7">TOPREAL3:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9" title="TOPREAL3:th.9">TOPREAL3:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9" title="TOPREAL3:th.9">TOPREAL3:9</a></i>
;<br/>
<a NAME="E25:53"/><i><font color="Green" title="E47">A20</font></i>: 
<font color="Maroon" title="c5">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E16:53"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_3"/>
( <font color="Maroon" title="c5">p'</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> or <font color="Maroon" title="c5">p'</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> or <font color="Maroon" title="c5">p'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> or <font color="Maroon" title="c5">p'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="sprect_3.html#E25:53"><i><font color="Green" title="E47">A20</font></i></a>, <a class="ref" href="sprect_3.html#T49" target="_self" title="SPRECT_3:th.49">Th49</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_3_1"><b>suppose </b></a><a NAME="E1:53_3_1"/>
<font color="Maroon" title="c5">p'</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:53_3_1"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="sprect_3.html#E8:53"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="sprect_3.html#E13:53"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="sprect_3.html#E20:53"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="sprect_3.html#E21:53"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="sprect_3.html#E23:53"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xreal_1.html#T182" title="XREAL_1:th.182">XREAL_1:182</a></i>;<br/><b>then </b><font color="Maroon" title="c5">p'</font> = 
<span class="p1">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E22:53"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
;<br/><b>hence </b><a NAME="E4:53_3_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_3_2"><b>suppose </b></a><a NAME="E1:53_3_2"/>
<font color="Maroon" title="c5">p'</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:53_3_2"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="sprect_3.html#E8:53"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="sprect_3.html#E13:53"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="sprect_3.html#E20:53"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="sprect_3.html#E21:53"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="sprect_3.html#E23:53"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="xreal_1.html#T181" title="XREAL_1:th.181">XREAL_1:181</a></i>;<br/><b>then </b><font color="Maroon" title="c5">p'</font> = 
<span class="p1">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E22:53"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
;<br/><b>hence </b><a NAME="E4:53_3_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_3_3"><b>suppose </b></a><a NAME="E1:53_3_3"/>
<font color="Maroon" title="c5">p'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:53_3_3"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="sprect_3.html#E9:53"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="sprect_3.html#E15:53"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="sprect_3.html#E20:53"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="sprect_3.html#E21:53"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="sprect_3.html#E24:53"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="xreal_1.html#T182" title="XREAL_1:th.182">XREAL_1:182</a></i>;<br/><b>then </b><font color="Maroon" title="c5">p'</font> = 
<span class="p1">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E22:53"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
;<br/><b>hence </b><a NAME="E4:53_3_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_3_4"><b>suppose </b></a><a NAME="E1:53_3_4"/>
<font color="Maroon" title="c5">p'</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:53_3_4"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="sprect_3.html#E9:53"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="sprect_3.html#E15:53"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="sprect_3.html#E20:53"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="sprect_3.html#E21:53"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="sprect_3.html#E24:53"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="xreal_1.html#T181" title="XREAL_1:th.181">XREAL_1:181</a></i>;<br/><b>then </b><font color="Maroon" title="c5">p'</font> = 
<span class="p1">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E22:53"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
;<br/><b>hence </b><a NAME="E4:53_3_4"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
