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<span class="kw">let </span><font color="Maroon" title="c1">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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">f</font> being  <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a><br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">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>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">j</font> &amp; <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">j</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">j</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">q</font>, <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> holds <br/><font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <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>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">j</font> &amp; <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">j</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>, <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">j</font> &amp; <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">j</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>, <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> implies <font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:6"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">j</font>
 <span class="kw">and </span><br/><a NAME="E2:6"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E3:6"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">j</font>
 <span class="kw">and </span><br/><a NAME="E4:6"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">j</font>
 <span class="kw">and </span><br/><a NAME="E5:6"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E6:6"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E7:6"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font>, <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/>

<a NAME="E8:6"/>
<font color="Maroon" title="c1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E9:6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A8</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:6_1"/>
( <font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1"><span class="kw">suppose </span></a><a NAME="E1:6_1_1"/><span class="lab"><font color="Green" title="E13">A9</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><div class="add"><a NAME="E2:6_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">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="#E3:6"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E6:6"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><a NAME="E3:6_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">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="#E1:6_1_1"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A10</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><span class="lab"><font color="Green" title="E15">A11</font></span>: 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1_1"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T121" title="XCMPLX_1:th.121">XCMPLX_1:121</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span>

;<br/><span class="lab"><font color="Green" title="E16">A12</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T7" title="TOPREAL3:th.7">TOPREAL3:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T9" title="TOPREAL3:th.9">TOPREAL3:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T9" title="TOPREAL3:th.9">TOPREAL3:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1_1"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></span>
<br/>.= 
<font color="Maroon" title="c4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 

;<br/><a NAME="E4:6_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">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="#E4:6"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E6:6"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><a NAME="E5:6_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A13</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E6:6_1_1_1"/><span class="lab"><font color="Green" title="E18">A14</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="goboard7.html#T6" title="GOBOARD7:th.6">GOBOARD7:6</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T7" title="TOPREAL3:th.7">TOPREAL3:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T9" title="TOPREAL3:th.9">TOPREAL3:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T9" title="TOPREAL3:th.9">TOPREAL3:9</a></span>
<br/>.= 
<font color="Maroon" title="c4">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="#E4:6"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="txt" href="#E6:6_1_1_1"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="ref" href="goboard7.html#T6" title="GOBOARD7:th.6">GOBOARD7:6</a></span>
;<br/><a NAME="E8:6_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A15</font></span>: 
<font color="Maroon" title="c4">q</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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_1_1_1"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></span>;<br/><span class="kw">assume </span><a NAME="E9:6_1_1_1"/><span class="lab"><font color="Green" title="E20">A16</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E10:6_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/><a NAME="E11:6_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1_1"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E2:6_1_1_1"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E12:6_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span>,<font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_1_1"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E5:6_1_1_1"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="txt" href="#E8:6_1_1_1"><span class="lab"><font color="Green" title="E19">A15</font></span></a></span>;<br/><a NAME="E13:6_1_1_1"/><span class="kw">then </span>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font>, <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="sprect_3.html#T40" title="SPRECT_3:th.40">SPRECT_3:40</a></span>;<br/><span class="kw">hence </span><a NAME="E14:6_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:6"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E9:6_1_1_1"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a>, <a class="ref" href="topreal1.html#T31" title="TOPREAL1:th.31">TOPREAL1:31</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">hence </span><a NAME="E6:6_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2"><span class="kw">suppose </span></a><a NAME="E1:6_1_2"/>
<font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:6_1_2"/>
<font color="Maroon" title="c3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">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="#E4:6"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E6:6"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
