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<span class="kw">let </span><font color="Maroon" title="c1">j</font>, <font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">k</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="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><br/> for <font color="Olive" title="b2">G</font> being  <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>  st <font color="Olive" title="b1">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Olive" title="b2">G</font> &amp;  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">f</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Olive" title="b2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Olive" title="b2">G</font> &amp; <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> holds <br/> ex <font color="Olive" title="b3">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <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">G</font> being  <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>  st <font color="Maroon" title="c4">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Olive" title="b1">G</font> &amp;  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Olive" title="b1">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> holds <br/> ex <font color="Olive" title="b2">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Olive" title="b1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</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="c4">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G</font> &amp;  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> implies  ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span> ) )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">X</font> = <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:4"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E2:4"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">and </span><br/><a NAME="E3:4"/><span class="lab"><font color="Green" title="E4">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E4:4"/><span class="lab"><font color="Green" title="E5">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E5:4"/><span class="lab"><font color="Green" title="E6">A5</font></span>: 
<font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span> )</span><br/>

<a NAME="E6:4"/><span class="lab"><font color="Green" title="E7">A6</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">by</span> <span class="lab"><a class="txt" href="#E3:4"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E7:4"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">X1</font> = <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pscomp_1.html#T64" title="PSCOMP_1:th.64">PSCOMP_1:64</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:4"/><span class="lab"><font color="Green" title="E8">A7</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">E-most</a> <font color="Maroon" title="c7">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c8">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:4"><span class="lab"><font color="Green" title="E8">A7</font></span></a></span>;<br/>
<a NAME="E12:4"/><span class="lab"><font color="Green" title="E9">A8</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:4"><span class="lab"><font color="Green" title="E8">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E13:4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A9</font></span>: 
<font color="Maroon" title="c9">p</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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E14:4"/>
<a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E11">A10</font></span>:  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span> = 
 <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p5">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>
<br/>.= 
 <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>

;<br/>
<a NAME="E16:4"/><span class="lab"><font color="Green" title="E12">A11</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:4"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<span class="lab"><font color="Green" title="E13">A12</font></span>: <font color="Maroon" title="c9">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:4"><span class="lab"><font color="Green" title="E8">A7</font></span></a>, <a class="ref" href="pscomp_1.html#T108" title="PSCOMP_1:th.108">PSCOMP_1:108</a></span>
<br/>.= 
 <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:4"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>
;<br/>
<a NAME="E18:4"/><span class="lab"><font color="Green" title="E14">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:4"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E19:4"/><span class="lab"><font color="Green" title="E15">A14</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:4"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E20:4"/><span class="lab"><font color="Green" title="E16">A15</font></span>: 
<font color="Maroon" title="c3">k</font> <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="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:4"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E21:4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="txt" href="#E6:4"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="ref" href="sprect_3.html#T25" title="SPRECT_3:th.25">SPRECT_3:25</a></span>;<br/>
<a NAME="E22:4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">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> <font color="Maroon" title="c9">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="#E13:4"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/>
<a NAME="E23:4"/><span class="lab"><font color="Green" title="E19">A18</font></span>: 
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:4"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="txt" href="#E21:4"><span class="lab"><font color="Green" title="E17">A16</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/>
<a NAME="E24:4"/><span class="lab"><font color="Green" title="E20">A19</font></span>: 
<font color="Maroon" title="c1">j</font> <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="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:4"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E21">A20</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">i</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="#E6:4"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</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="#E18:4"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E20:4"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E16:4"><span class="lab"><font color="Green" title="E12">A11</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
;<br/>
<a NAME="E26:4"/>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:4"><span class="lab"><font color="Green" title="E9">A8</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E27:4"/><span class="kw">then </span>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">k1</font></span>)</span> where <font color="Olive" title="b1">k1</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k1</font> &amp; <font color="Olive" title="b1">k1</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="c4">f</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E29:4"/><span class="lab"><font color="Green" title="E22">A21</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font>
 <span class="kw">and </span><br/><a NAME="E30:4"/><span class="lab"><font color="Green" title="E23">A22</font></span>: 
<font color="Maroon" title="c10">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">k1</font></span>)</span> where <font color="Olive" title="b1">k1</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k1</font> &amp; <font color="Olive" title="b1">k1</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="c4">f</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">i1</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E32:4"/><span class="lab"><font color="Green" title="E24">A23</font></span>: 
<font color="Maroon" title="c10">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c11">i1</font>
 <span class="kw">and </span><br/><a NAME="E33:4"/><span class="lab"><font color="Green" title="E25">A24</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i1</font>
 <span class="kw">and </span><br/><a NAME="E34:4"/><span class="lab"><font color="Green" title="E26">A25</font></span>: 
<font color="Maroon" title="c11">i1</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="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:4"><span class="lab"><font color="Green" title="E23">A22</font></span></a></span>;<br/>
<a NAME="E35:4"/><span class="lab"><font color="Green" title="E27">A26</font></span>: 
<font color="Maroon" title="c9">p</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="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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="#E29:4"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="txt" href="#E32:4"><span class="lab"><font color="Green" title="E24">A23</font></span></a>, <a class="txt" href="#E33:4"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="txt" href="#E34:4"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></span>;<br/>
<a NAME="E36:4"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:4"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E37:4"/><span class="kw">then </span>
<font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:4"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E38:4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A27</font></span>: 
<font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">jo</font>, <font color="Maroon" title="c13">io</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E40:4"/><span class="lab"><font color="Green" title="E29">A28</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">jo</font>,<font color="Maroon" title="c13">io</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E41:4"/><span class="lab"><font color="Green" title="E30">A29</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">jo</font>,<font color="Maroon" title="c13">io</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/>
<a NAME="E42:4"/><span class="lab"><font color="Green" title="E31">A30</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">jo</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:4"><span class="lab"><font color="Green" title="E29">A28</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E43:4"/>
<font color="Maroon" title="c11">i1</font> <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="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:4"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E44:4"/><span class="kw">then </span>
<font color="Maroon" title="c11">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:4"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E45:4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A31</font></span>: 
<font color="Maroon" title="c11">i1</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> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">j0</font>, <font color="Maroon" title="c15">i0</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E47:4"/><span class="lab"><font color="Green" title="E33">A32</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">j0</font>,<font color="Maroon" title="c15">i0</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E48:4"/><span class="lab"><font color="Green" title="E34">A33</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">j0</font>,<font color="Maroon" title="c15">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/>
<a NAME="E49:4"/><span class="lab"><font color="Green" title="E35">A34</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:4"><span class="lab"><font color="Green" title="E33">A32</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E50:4"/><span class="lab"><font color="Green" title="E36">A35</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i0</font> &amp; <font color="Maroon" title="c15">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:4"><span class="lab"><font color="Green" title="E33">A32</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E51:4"/><span class="lab"><font color="Green" title="E37">A36</font></span>: 
<font color="Maroon" title="c14">j0</font> <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="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:4"><span class="lab"><font color="Green" title="E33">A32</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E52:4"/><span class="lab"><font color="Green" title="E38">A37</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">io</font> &amp; <font color="Maroon" title="c13">io</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:4"><span class="lab"><font color="Green" title="E29">A28</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E53:4"/><span class="lab"><font color="Green" title="E39">A38</font></span>: 
<font color="Maroon" title="c12">jo</font> <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="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:4"><span class="lab"><font color="Green" title="E29">A28</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E54:4"/><span class="lab"><font color="Green" title="E40">A39</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E45:4"><span class="lab"><font color="Green" title="E32">A31</font></span></a>, <a class="ref" href="jordan8.html#T7" title="JORDAN8:th.7">JORDAN8:7</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></span>;<br/>
<a NAME="E55:4"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_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="E1:4_4_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</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="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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>  or <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:4"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="txt" href="#E34:4"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="txt" href="#E54:4"><span class="lab"><font color="Green" title="E40">A39</font></span></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:4_4_1_1"/><span class="lab"><font color="Green" title="E41">A40</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</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="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c13">io</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c13">io</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="#E6:4"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E24:4"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E52:4"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">jo</font>,<font color="Maroon" title="c13">io</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="#E42:4"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="txt" href="#E53:4"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="txt" href="#E52:4"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
<br/>.= 
<font color="Maroon" title="c9">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="#E35:4"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="txt" href="#E41:4"><span class="lab"><font color="Green" title="E30">A29</font></span></a>, <a class="txt" href="#E1:4_4_1_1"><span class="lab"><font color="Green" title="E41">A40</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"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</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="#E25:4"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E13:4"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="ref" href="goboard7.html#T6" title="GOBOARD7:th.6">GOBOARD7:6</a></span>
;<br/><a NAME="E3:4_4_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c13">io</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c13">io</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E24:4"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E52:4"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></span>;<br/><a NAME="E4:4_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A42</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">io</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c15">i0</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c15">i0</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="#E6:4"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E24:4"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E50:4"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">j0</font>,<font color="Maroon" title="c15">i0</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="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="txt" href="#E50:4"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
<br/>.= 
<font color="Maroon" title="c9">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="#E35:4"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="txt" href="#E48:4"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E1:4_4_1_1"><span class="lab"><font color="Green" title="E41">A40</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"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</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="#E25:4"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E13:4"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="ref" href="goboard7.html#T6" title="GOBOARD7:th.6">GOBOARD7:6</a></span>
;<br/><a NAME="E6:4_4_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c15">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c15">i0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E24:4"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E50:4"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></span>;<br/><a NAME="E7:4_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A44</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">thus </span><a NAME="E8:4_4_1_1"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_1_1_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_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="E1:4_4_1_1_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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>  or <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</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="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:4_4_1_1_3_1_1"/><span class="lab"><font color="Green" title="E44">A45</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:4_4_1_1_3_1_1"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_1_1_3_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:4_4_1_1_3_1_1_1_1"/>
( <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> or  not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:4_4_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A46</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><div class="add"><a NAME="E2:4_4_1_1_3_1_1_1_1_1"/>
1 <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> <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:4"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E3:4_4_1_1_3_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:4"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="txt" href="#E38:4"><span class="lab"><font color="Green" title="E28">A27</font></span></a>, <a class="ref" href="goboard1.html#T16" title="GOBOARD1:th.16">GOBOARD1:16</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:4_4_1_1_3_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4_4_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:4_4_1_1_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A47</font></span>: 
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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">by</span> <span class="lab"><a class="txt" href="#E15:4"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="txt" href="#E17:4"><span class="lab"><font color="Green" title="E13">A12</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c16">n</font> = <font color="Maroon" title="c12">jo</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font> &amp; <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><a NAME="E6:4_4_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A48</font></span>: 
<font color="Maroon" title="c9">p</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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:4"><span class="lab"><font color="Green" title="E9">A8</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E7:4_4_1_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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">by</span> <span class="lab"><a class="txt" href="#E35:4"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><a NAME="E8:4_4_1_1_3_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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">by</span> <span class="lab"><a class="txt" href="#E5:4_4_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A49</font></span>: <font color="Maroon" title="c9">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">jo</font>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:4"><span class="lab"><font color="Green" title="E30">A29</font></span></a>, <a class="txt" href="#E42:4"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="txt" href="#E53:4"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="txt" href="#E52:4"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E42:4"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="txt" href="#E53:4"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>
;<br/><a NAME="E10:4_4_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A50</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="txt" href="#E6:4"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E20:4"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="ref" href="sprect_3.html#T25" title="SPRECT_3:th.25">SPRECT_3:25</a></span>;<br/><a NAME="E11:4_4_1_1_3_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4_4_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A48</font></span></a>, <a class="txt" href="#E9:4_4_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A49</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E12:4_4_1_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:4"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E42:4"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></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="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><a NAME="E13:4_4_1_1_3_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4_4_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A48</font></span></a>, <a class="txt" href="#E9:4_4_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A49</font></span></a>, <a class="txt" href="#E10:4_4_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E14:4_4_1_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:4"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E16:4"><span class="lab"><font color="Green" title="E12">A11</font></span></a>, <a class="txt" href="#E53:4"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font></span><br/><font color="Maroon" title="c9">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</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="#E25:4"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E6:4_4_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A48</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"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">i</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="#E6:4"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E24:4"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</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="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E42:4"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="txt" href="#E53:4"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
;<br/><span class="kw">hence </span><a NAME="E16:4_4_1_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:4_4_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A49</font></span></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:4_4_1_1_3_1_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A51</font></span>: 
not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><div class="add"><span class="lab"><font color="Green" title="E46">A52</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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>  = 
<font color="Maroon" title="c9">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="#E35:4"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="txt" href="#E1:4_4_1_1"><span class="lab"><font color="Green" title="E41">A40</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"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</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="#E25:4"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E13:4"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="ref" href="goboard7.html#T6" title="GOBOARD7:th.6">GOBOARD7:6</a></span>
;<br/><span class="kw">thus </span><a NAME="E3:4_4_1_1_3_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_1_1_3_1_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_1_1_1_2_2_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="E1:4_4_1_1_3_1_1_1_1_2_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:4"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E45">A51</font></span></a>, <a class="txt" href="#E2:4_4_1_1_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">A52</font></span></a>, <a class="ref" href="goboard7.html#T9" title="GOBOARD7:th.9">GOBOARD7:9</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_1_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E47">A53</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</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">  ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><div class="add"><a NAME="E2:4_4_1_1_3_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">j0</font>,<font color="Maroon" title="c15">i0</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:4"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="txt" href="#E48:4"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><a NAME="E3:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">j0</font>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="txt" href="#E50:4"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>;<br/><a NAME="E4:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">j0</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>;<br/><a NAME="E5:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">j0</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:4"><span class="lab"><font color="Green" title="E19">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A54</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c14">j0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:4"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E16:4"><span class="lab"><font color="Green" title="E12">A11</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>;<br/><a NAME="E7:4_4_1_1_3_1_1_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">i0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">io</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">j0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">jo</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E45:4"><span class="lab"><font color="Green" title="E32">A31</font></span></a>, <a class="txt" href="#E38:4"><span class="lab"><font color="Green" title="E28">A27</font></span></a>, <a class="txt" href="#E47:4"><span class="lab"><font color="Green" title="E33">A32</font></span></a>, <a class="txt" href="#E48:4"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E40:4"><span class="lab"><font color="Green" title="E29">A28</font></span></a>, <a class="txt" href="#E41:4"><span class="lab"><font color="Green" title="E30">A29</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><a NAME="E8:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">j0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">jo</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:4_4_1_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E4:4_4_1_1"><span class="lab"><font color="Green" title="E42">A42</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>;<br/><a NAME="E9:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A55</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">j0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">jo</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>;<br/><a NAME="E10:4_4_1_1_3_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c14">j0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">jo</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:4"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E41:4"><span class="lab"><font color="Green" title="E30">A29</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="txt" href="#E50:4"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="txt" href="#E42:4"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="txt" href="#E7:4_4_1_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E4:4_4_1_1"><span class="lab"><font color="Green" title="E42">A42</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>;<br/><a NAME="E11:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">j0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">jo</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/><a NAME="E12:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">jo</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:4_4_1_1_3_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E49">A55</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E13:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A56</font></span>: 
<font color="Maroon" title="c14">j0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">jo</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E14:4_4_1_1_3_1_1_1_1_2_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">jo</font>,<font color="Maroon" title="c2">i</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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c12">jo</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:4"><span class="lab"><font color="Green" title="E30">A29</font></span></a>, <a class="txt" href="#E20:4"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E16:4"><span class="lab"><font color="Green" title="E12">A11</font></span></a>, <a class="txt" href="#E42:4"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="txt" href="#E4:4_4_1_1"><span class="lab"><font color="Green" title="E42">A42</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E47">A53</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>;<br/><a NAME="E15:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">jo</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E16:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">j0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:4_4_1_1_3_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E50">A56</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E17:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A57</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4_4_1_1_3_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E48">A54</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c16">n</font> = <font color="Maroon" title="c14">j0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font> &amp; <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><span class="lab"><font color="Green" title="E52">A58</font></span>: <font color="Maroon" title="c9">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</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="#E25:4"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E13:4"><span class="lab"><font color="Green" title="E10">A9</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"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">i</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="#E6:4"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E24:4"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</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="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
;<br/><a NAME="E19:4_4_1_1_3_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">j0</font>,<font color="Maroon" title="c15">i0</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:4"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="txt" href="#E48:4"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><a NAME="E20:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">j0</font>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="txt" href="#E50:4"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>;<br/><a NAME="E21:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">j0</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>;<br/><a NAME="E22:4_4_1_1_3_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:4"><span class="lab"><font color="Green" title="E19">A18</font></span></a>, <a class="txt" href="#E17:4_4_1_1_3_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E51">A57</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E23:4_4_1_1_3_1_1_1_1_2_2_1_1"/>
( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font> &amp; <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="txt" href="#E17:4_4_1_1_3_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E51">A57</font></span></a>, <a class="txt" href="#E18:4_4_1_1_3_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E52">A58</font></span></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_1_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:4_4_1_1_3_1_1_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E47">A59</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</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">  ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><div class="add"><a NAME="E2:4_4_1_1_3_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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">by</span> <span class="lab"><a class="txt" href="#E35:4"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E3:4_4_1_1_3_1_1_1_1_2_2_1_2"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:4"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_1_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E47">A59</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:4_4_1_1_3_1_2"/><span class="lab"><font color="Green" title="E44">A60</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</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="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:4_4_1_1_3_1_2"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_1_1_3_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:4_4_1_1_3_1_2_1_1"/>
( <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> or  not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:4_4_1_1_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E45">A61</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><div class="add"><a NAME="E2:4_4_1_1_3_1_2_1_1_1"/>
1 <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> <font color="Maroon" title="c11">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:4"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E3:4_4_1_1_3_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:4"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="txt" href="#E45:4"><span class="lab"><font color="Green" title="E32">A31</font></span></a>, <a class="ref" href="goboard1.html#T16" title="GOBOARD1:th.16">GOBOARD1:16</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:4_4_1_1_3_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4_4_1_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E45">A61</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:4_4_1_1_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A62</font></span>: 
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:4"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="txt" href="#E17:4"><span class="lab"><font color="Green" title="E13">A12</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c16">n</font> = <font color="Maroon" title="c14">j0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font> &amp; <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><a NAME="E6:4_4_1_1_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E47">A63</font></span>: 
<font color="Maroon" title="c9">p</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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:4"><span class="lab"><font color="Green" title="E9">A8</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E7:4_4_1_1_3_1_2_1_1_1"/>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:4"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_2"><span class="lab"><font color="Green" title="E44">A60</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><a NAME="E8:4_4_1_1_3_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4_4_1_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E46">A62</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A64</font></span>: <font color="Maroon" title="c9">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">j0</font>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:4"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="txt" href="#E50:4"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>
;<br/><a NAME="E10:4_4_1_1_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E49">A65</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="txt" href="#E6:4"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E20:4"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="ref" href="sprect_3.html#T25" title="SPRECT_3:th.25">SPRECT_3:25</a></span>;<br/><a NAME="E11:4_4_1_1_3_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4_4_1_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E47">A63</font></span></a>, <a class="txt" href="#E9:4_4_1_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E48">A64</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E12:4_4_1_1_3_1_2_1_1_1"/>
<font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:4"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></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="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><a NAME="E13:4_4_1_1_3_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4_4_1_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E47">A63</font></span></a>, <a class="txt" href="#E9:4_4_1_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E48">A64</font></span></a>, <a class="txt" href="#E10:4_4_1_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E49">A65</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E14:4_4_1_1_3_1_2_1_1_1"/>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:4"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E16:4"><span class="lab"><font color="Green" title="E12">A11</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font></span><br/><font color="Maroon" title="c9">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</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="#E25:4"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E6:4_4_1_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E47">A63</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"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">i</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="#E6:4"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E24:4"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</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="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
;<br/><span class="kw">hence </span><a NAME="E16:4_4_1_1_3_1_2_1_1_1"/>
<font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:4_4_1_1_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E48">A64</font></span></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:4_4_1_1_3_1_2_1_1_2"/><span class="lab"><font color="Green" title="E45">A66</font></span>: 
not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><div class="add"><span class="lab"><font color="Green" title="E46">A67</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<font color="Maroon" title="c9">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="#E35:4"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="txt" href="#E1:4_4_1_1"><span class="lab"><font color="Green" title="E41">A40</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"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</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="#E25:4"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E13:4"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="ref" href="goboard7.html#T6" title="GOBOARD7:th.6">GOBOARD7:6</a></span>
;<br/><span class="kw">thus </span><a NAME="E3:4_4_1_1_3_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_4_1_1_3_1_2_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_2_1_1_2_2_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="E1:4_4_1_1_3_1_2_1_1_2_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:4"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E45">A66</font></span></a>, <a class="txt" href="#E2:4_4_1_1_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E46">A67</font></span></a>, <a class="ref" href="goboard7.html#T9" title="GOBOARD7:th.9">GOBOARD7:9</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_2_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E47">A68</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</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">  ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><div class="add"><a NAME="E2:4_4_1_1_3_1_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">jo</font>,<font color="Maroon" title="c13">io</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:4"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="txt" href="#E41:4"><span class="lab"><font color="Green" title="E30">A29</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_2"><span class="lab"><font color="Green" title="E44">A60</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><a NAME="E3:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">jo</font>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:4"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="txt" href="#E53:4"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="txt" href="#E52:4"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>;<br/><a NAME="E4:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">jo</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E42:4"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="txt" href="#E53:4"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>;<br/><a NAME="E5:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">jo</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:4"><span class="lab"><font color="Green" title="E19">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A69</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">jo</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:4"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E16:4"><span class="lab"><font color="Green" title="E12">A11</font></span></a>, <a class="txt" href="#E53:4"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>;<br/><a NAME="E7:4_4_1_1_3_1_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c12">jo</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:4"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E41:4"><span class="lab"><font color="Green" title="E30">A29</font></span></a>, <a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E50:4"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="txt" href="#E53:4"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="txt" href="#E7:4_4_1_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E4:4_4_1_1"><span class="lab"><font color="Green" title="E42">A42</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_2"><span class="lab"><font color="Green" title="E44">A60</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>;<br/><a NAME="E8:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">jo</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/><a NAME="E9:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A70</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">jo</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j0</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E10:4_4_1_1_3_1_2_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">i0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">io</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">j0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">jo</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E45:4"><span class="lab"><font color="Green" title="E32">A31</font></span></a>, <a class="txt" href="#E38:4"><span class="lab"><font color="Green" title="E28">A27</font></span></a>, <a class="txt" href="#E47:4"><span class="lab"><font color="Green" title="E33">A32</font></span></a>, <a class="txt" href="#E48:4"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E40:4"><span class="lab"><font color="Green" title="E29">A28</font></span></a>, <a class="txt" href="#E41:4"><span class="lab"><font color="Green" title="E30">A29</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><a NAME="E11:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">j0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">jo</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:4_4_1_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E4:4_4_1_1"><span class="lab"><font color="Green" title="E42">A42</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>;<br/><a NAME="E12:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">j0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">jo</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:4_4_1_1_3_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E49">A70</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E13:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A71</font></span>: 
<font color="Maroon" title="c12">jo</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E14:4_4_1_1_3_1_2_1_1_2_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">j0</font>,<font color="Maroon" title="c2">i</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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c14">j0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:4"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E20:4"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E16:4"><span class="lab"><font color="Green" title="E12">A11</font></span></a>, <a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E7:4_4_1_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E47">A68</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></span>;<br/><a NAME="E15:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">j0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E16:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">jo</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:4_4_1_1_3_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E50">A71</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E17:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A72</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">jo</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4_4_1_1_3_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E48">A69</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c16">n</font> = <font color="Maroon" title="c12">jo</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font> &amp; <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><span class="lab"><font color="Green" title="E52">A73</font></span>: <font color="Maroon" title="c9">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</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="#E25:4"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E13:4"><span class="lab"><font color="Green" title="E10">A9</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"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">i</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="#E6:4"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E24:4"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</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="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E42:4"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="txt" href="#E53:4"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
;<br/><a NAME="E19:4_4_1_1_3_1_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">jo</font>,<font color="Maroon" title="c13">io</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:4"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="txt" href="#E41:4"><span class="lab"><font color="Green" title="E30">A29</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_2"><span class="lab"><font color="Green" title="E44">A60</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><a NAME="E20:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">jo</font>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:4"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="txt" href="#E53:4"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="txt" href="#E52:4"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>;<br/><a NAME="E21:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">jo</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E42:4"><span class="lab"><font color="Green" title="E31">A30</font></span></a>, <a class="txt" href="#E53:4"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>;<br/><a NAME="E22:4_4_1_1_3_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:4"><span class="lab"><font color="Green" title="E19">A18</font></span></a>, <a class="txt" href="#E17:4_4_1_1_3_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E51">A72</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E23:4_4_1_1_3_1_2_1_1_2_2_1_1"/>
( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font> &amp; <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="txt" href="#E17:4_4_1_1_3_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E51">A72</font></span></a>, <a class="txt" href="#E18:4_4_1_1_3_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E52">A73</font></span></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_1_3_1_2_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:4_4_1_1_3_1_2_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E47">A74</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</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">  ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><div class="add"><a NAME="E2:4_4_1_1_3_1_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:4"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_2"><span class="lab"><font color="Green" title="E44">A60</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E3:4_4_1_1_3_1_2_1_1_2_2_1_2"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:4"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E1:4_4_1_1_3_1_2_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E47">A74</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:4_4_1_2"/><span class="lab"><font color="Green" title="E41">A75</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i1</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c16">n</font> = <font color="Maroon" title="c14">j0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font> &amp; <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><a NAME="E2:4_4_1_2"/>
<font color="Maroon" title="c9">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="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:4"><span class="lab"><font color="Green" title="E27">A26</font></span></a>, <a class="txt" href="#E1:4_4_1_2"><span class="lab"><font color="Green" title="E41">A75</font></span></a>, <a class="ref" href="goboard7.html#T5" title="GOBOARD7:th.5">GOBOARD7:5</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A76</font></span>: <font color="Maroon" title="c9">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">j0</font>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:4"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="txt" href="#E50:4"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></span>
;<br/><a NAME="E4:4_4_1_2"/><span class="lab"><font color="Green" title="E43">A77</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="txt" href="#E6:4"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E20:4"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="ref" href="sprect_3.html#T25" title="SPRECT_3:th.25">SPRECT_3:25</a></span>;<br/><a NAME="E5:4_4_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:4"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="txt" href="#E3:4_4_1_2"><span class="lab"><font color="Green" title="E42">A76</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E6:4_4_1_2"/>
<font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:4"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></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="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )</span><br/><a NAME="E7:4_4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:4"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="txt" href="#E3:4_4_1_2"><span class="lab"><font color="Green" title="E42">A76</font></span></a>, <a class="txt" href="#E4:4_4_1_2"><span class="lab"><font color="Green" title="E43">A77</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E8:4_4_1_2"/>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:4"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E16:4"><span class="lab"><font color="Green" title="E12">A11</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></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="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font></span><br/><font color="Maroon" title="c9">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</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="#E25:4"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E13:4"><span class="lab"><font color="Green" title="E10">A9</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"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">i</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="#E6:4"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E24:4"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="txt" href="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</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="#E19:4"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E49:4"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E51:4"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:4_4_1_2"/>
<font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">n</font>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:4_4_1_2"><span class="lab"><font color="Green" title="E42">A76</font></span></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E56:4"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st <br/>( <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a>  <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">j</font>,<font color="Maroon" title="c2">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:4"><span class="lab"><font color="Green" title="E13">A12</font></span></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>
