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<span class="kw">let </span><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">f2</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 2 <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="c1">f1</font> &amp; 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font> &amp; <font color="Maroon" title="c1">f1</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp; <font color="Maroon" title="c2">f2</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</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="c1">f1</font> &amp; <font color="Olive" title="b1">n</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="c1">f1</font> holds <br/><font color="Maroon" title="c1">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</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="c2">f2</font> &amp; <font color="Olive" title="b1">n</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="c2">f2</font> holds <br/><font color="Maroon" title="c2">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) &amp;  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c1">f1</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font></span>)</span> &amp;  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c2">f2</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font></span>)</span> &amp;  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c1">f1</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font></span>)</span> &amp;  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c2">f2</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font></span>)</span> implies  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f1</font> <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="c2">f2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:5"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
2 <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="c1">f1</font>
 <span class="kw">and </span><br/><a NAME="E2:5"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font>
 <span class="kw">and </span><br/><a NAME="E3:5"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c1">f1</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">and </span><br/><a NAME="E4:5"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c2">f2</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">and </span><br/><a NAME="E5:5"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</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="c1">f1</font> &amp; <font color="Olive" title="b1">n</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="c1">f1</font> holds <br/><font color="Maroon" title="c1">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:5"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</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="c2">f2</font> &amp; <font color="Olive" title="b1">n</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="c2">f2</font> holds <br/><font color="Maroon" title="c2">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:5"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c1">f1</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:5"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c2">f2</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:5"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c1">f1</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:5"/><span class="lab"><font color="Green" title="E15">A10</font></span>: 
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c2">f2</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f1</font> <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="c2">f2</font></span><br/>

<a NAME="E11:5"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E7">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">f1</font> = <font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c4">f2</font> = <font color="Maroon" title="c2">f2</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">f2</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E14:5"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
 <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">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f2</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">reconsider </span><font color="Maroon" title="c6">p1</font> = <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1, <font color="Maroon" title="c7">p2</font> = <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span>, <font color="Maroon" title="c8">q1</font> = <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1, <font color="Maroon" title="c9">q2</font> = <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</font></span>)</span> 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> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">x</font> =  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">y</font> =  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">x1</font> =  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">y1</font> =  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">x2</font> =  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c4">f2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">y2</font> =  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c4">f2</font>;<br/>
<a NAME="E16:5"/><span class="lab"><font color="Green" title="E17">A12</font></span>: 
 <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="c3">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f1</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/>
<a NAME="E17:5"/><span class="lab"><font color="Green" title="E18">A13</font></span>: 
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="c3">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E18:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A14</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E19:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A15</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/>
<a NAME="E20:5"/><span class="lab"><font color="Green" title="E21">A16</font></span>: 
 <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="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">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">set </span><font color="Maroon" title="c16">G</font> =  <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font>;<br/>
<a NAME="E21:5"/><span class="lab"><font color="Green" title="E22">A17</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/>
<a NAME="E22:5"/><span class="lab"><font color="Green" title="E23">A18</font></span>: 
(  <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> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c4">f2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c4">f2</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</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>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/>
<a NAME="E23:5"/><span class="lab"><font color="Green" title="E24">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E24:5"/><span class="lab"><font color="Green" title="E25">A20</font></span>: 
(  <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> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</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>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/>
<a NAME="E25:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A21</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:5"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E18:5"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E27">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">m</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="c5">f</font> implies <font color="Maroon" title="c6">p1</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"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c18">s</font> = <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font>;<br/><span class="kw">assume </span><a NAME="E1:5_1"/><span class="lab"><font color="Green" title="E28">A23</font></span>: 
<font color="Maroon" title="c17">m</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="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">p1</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"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font></span><br/><a NAME="E2:5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A24</font></span>: 
<font color="Maroon" title="c17">m</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">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:5_1"/><span class="lab"><font color="Green" title="E30">A25</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_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="E1:5_1_1_1"/>
( <font color="Maroon" title="c17">m</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="c3">f1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:5_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A26</font></span>: 
<font color="Maroon" title="c17">m</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="c3">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">p1</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"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c19">u</font> = <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">m</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> ;<br/><a NAME="E3:5_1_1_1_1"/><span class="lab"><font color="Green" title="E32">A27</font></span>: 
<font color="Maroon" title="c17">m</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="c3">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_1"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="txt" href="#E1:5_1_1_1_1"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:5_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E5:5_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A28</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:5"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E21:5"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="txt" href="#E1:5_1"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/><a NAME="E6:5_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:5"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E24:5"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="txt" href="#E3:5_1_1_1_1"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:5_1_1_1_1"/>
<font color="Maroon" title="c6">p1</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"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:5"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E16:5"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E24:5"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="txt" href="#E25:5"><span class="lab"><font color="Green" title="E26">A21</font></span></a>, <a class="txt" href="#E3:5_1_1_1_1"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E5:5_1_1_1_1"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="ref" href="goboard4.html#D2" target="_self" title="GOBOARD4:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="5_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:5_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A29</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">p1</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"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">w5</font> = <font color="Maroon" title="c17">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/><a NAME="E3:5_1_1_1_2"/>
<font color="Maroon" title="c19">w5</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1_1_1_2"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E4:5_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A30</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">w5</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E5:5_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A31</font></span>: 
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">w5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">m1</font> = <font color="Maroon" title="c17">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">u</font> = <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">m1</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> ;<br/><a NAME="E8:5_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A32</font></span>: 
<font color="Maroon" title="c19">w5</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">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:5"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="txt" href="#E2:5_1"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="txt" href="#E5:5_1_1_1_2"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E9:5_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">w5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:5_1_1_1_2"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="txt" href="#E4:5_1_1_1_2"><span class="lab"><font color="Green" title="E32">A30</font></span></a>, <a class="ref" href="seq_4.html#T136" title="SEQ_4:th.136">SEQ_4:136</a></span>;<br/><a NAME="E10:5_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A33</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:5"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E21:5"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="txt" href="#E1:5_1"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/><a NAME="E11:5_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A34</font></span>: 
<font color="Maroon" title="c20">m1</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">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_1_1_1_2"><span class="lab"><font color="Green" title="E32">A30</font></span></a>, <a class="txt" href="#E8:5_1_1_1_2"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E12:5_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E22:5"><span class="lab"><font color="Green" title="E23">A18</font></span></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E13:5_1_1_1_2"/>
<font color="Maroon" title="c6">p1</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"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:5"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E22:5"><span class="lab"><font color="Green" title="E23">A18</font></span></a>, <a class="txt" href="#E25:5"><span class="lab"><font color="Green" title="E26">A21</font></span></a>, <a class="txt" href="#E11:5_1_1_1_2"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="txt" href="#E10:5_1_1_1_2"><span class="lab"><font color="Green" title="E35">A33</font></span></a>, <a class="ref" href="goboard4.html#D2" target="_self" title="GOBOARD4:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E5:5_1"/>
<font color="Maroon" title="c6">p1</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"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E27:5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E28:5"/><span class="kw">then </span>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E29:5"/><span class="kw">then </span>
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="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E30:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A35</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E31:5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:5"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E21:5"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="txt" href="#E19:5"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/>
<a NAME="E32:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A36</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:5"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="txt" href="#E26:5"><span class="lab"><font color="Green" title="E27">A22</font></span></a>, <a class="ref" href="goboard2.html#T15" title="GOBOARD2:th.15">GOBOARD2:15</a></span>;<br/>
<a NAME="E33:5"/><span class="lab"><font color="Green" title="E30">A37</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</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="c3">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:5"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E34:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A38</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/>
<a NAME="E35:5"/><span class="lab"><font color="Green" title="E32">A39</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:5"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E24:5"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="txt" href="#E33:5"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E33">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">m</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="c5">f</font> implies <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/><span class="kw">set </span><font color="Maroon" title="c18">s</font> = <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font>;<br/><span class="kw">assume </span><a NAME="E1:5_2"/><span class="lab"><font color="Green" title="E34">A41</font></span>: 
<font color="Maroon" title="c17">m</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="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><a NAME="E2:5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A42</font></span>: 
<font color="Maroon" title="c17">m</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">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:5_2"/><span class="lab"><font color="Green" title="E36">A43</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_2"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_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="E1:5_2_1_1"/>
( <font color="Maroon" title="c17">m</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="c3">f1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:5_2_1_1_1"/><span class="lab"><font color="Green" title="E37">A44</font></span>: 
<font color="Maroon" title="c17">m</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="c3">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c19">u</font> = <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">m</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> ;<br/><a NAME="E3:5_2_1_1_1"/><span class="lab"><font color="Green" title="E38">A45</font></span>: 
<font color="Maroon" title="c17">m</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="c3">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_2"><span class="lab"><font color="Green" title="E36">A43</font></span></a>, <a class="txt" href="#E1:5_2_1_1_1"><span class="lab"><font color="Green" title="E37">A44</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:5_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E5:5_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A46</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:5"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E21:5"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="txt" href="#E1:5_2"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/><a NAME="E6:5_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:5"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E24:5"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="txt" href="#E3:5_2_1_1_1"><span class="lab"><font color="Green" title="E38">A45</font></span></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:5_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:5"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E16:5"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E24:5"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="txt" href="#E35:5"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E3:5_2_1_1_1"><span class="lab"><font color="Green" title="E38">A45</font></span></a>, <a class="txt" href="#E5:5_2_1_1_1"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="ref" href="goboard4.html#D2" target="_self" title="GOBOARD4:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="5_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:5_2_1_1_2"/><span class="lab"><font color="Green" title="E37">A47</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">w5</font> = <font color="Maroon" title="c17">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/><a NAME="E3:5_2_1_1_2"/>
<font color="Maroon" title="c19">w5</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_2_1_1_2"><span class="lab"><font color="Green" title="E37">A47</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E4:5_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A48</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">w5</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E5:5_2_1_1_2"/><span class="lab"><font color="Green" title="E39">A49</font></span>: 
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">w5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">m1</font> = <font color="Maroon" title="c17">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">u</font> = <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">m1</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> ;<br/><a NAME="E8:5_2_1_1_2"/><span class="lab"><font color="Green" title="E40">A50</font></span>: 
<font color="Maroon" title="c19">w5</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">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:5"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="txt" href="#E2:5_2"><span class="lab"><font color="Green" title="E35">A42</font></span></a>, <a class="txt" href="#E5:5_2_1_1_2"><span class="lab"><font color="Green" title="E39">A49</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E9:5_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">w5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:5_2_1_1_2"><span class="lab"><font color="Green" title="E39">A49</font></span></a>, <a class="txt" href="#E4:5_2_1_1_2"><span class="lab"><font color="Green" title="E38">A48</font></span></a>, <a class="ref" href="seq_4.html#T136" title="SEQ_4:th.136">SEQ_4:136</a></span>;<br/><a NAME="E10:5_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A51</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:5"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E21:5"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="txt" href="#E1:5_2"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/><a NAME="E11:5_2_1_1_2"/><span class="lab"><font color="Green" title="E42">A52</font></span>: 
<font color="Maroon" title="c20">m1</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">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_2_1_1_2"><span class="lab"><font color="Green" title="E38">A48</font></span></a>, <a class="txt" href="#E8:5_2_1_1_2"><span class="lab"><font color="Green" title="E40">A50</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E12:5_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E22:5"><span class="lab"><font color="Green" title="E23">A18</font></span></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E13:5_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:5"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E22:5"><span class="lab"><font color="Green" title="E23">A18</font></span></a>, <a class="txt" href="#E35:5"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E11:5_2_1_1_2"><span class="lab"><font color="Green" title="E42">A52</font></span></a>, <a class="txt" href="#E10:5_2_1_1_2"><span class="lab"><font color="Green" title="E41">A51</font></span></a>, <a class="ref" href="goboard4.html#D2" target="_self" title="GOBOARD4:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E5:5_2"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E37:5"/><span class="lab"><font color="Green" title="E34">A53</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T26" title="FINSEQ_1:th.26">FINSEQ_1:26</a></span>;<br/>
<a NAME="E38:5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:5"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E21:5"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="txt" href="#E33:5"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E34:5"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/>
<a NAME="E39:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A54</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> (<span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:5"><span class="lab"><font color="Green" title="E34">A53</font></span></a>, <a class="txt" href="#E33:5"><span class="lab"><font color="Green" title="E30">A37</font></span></a>, <a class="txt" href="#E36:5"><span class="lab"><font color="Green" title="E33">A40</font></span></a>, <a class="ref" href="goboard2.html#T16" title="GOBOARD2:th.16">GOBOARD2:16</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">n</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="c3">f1</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) ) )</span><br/><span class="kw">assume </span><a NAME="E1:5_3"/><span class="lab"><font color="Green" title="E36">A55</font></span>: 
<font color="Maroon" title="c17">n</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="c3">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )</span><br/><a NAME="E2:5_3"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T26" title="FINSEQ_1:th.26">FINSEQ_1:26</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">i</font>, <font color="Maroon" title="c19">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:5_3"/><span class="lab"><font color="Green" title="E37">A56</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">j</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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_3"><span class="lab"><font color="Green" title="E36">A55</font></span></a>, <a class="ref" href="goboard2.html#T14" title="GOBOARD2:th.14">GOBOARD2:14</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c20">i</font> = <font color="Maroon" title="c18">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i</font>,<font color="Olive" title="b1">j</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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c20">i</font>,<font color="Olive" title="b1">j</font>) )</span><br/><span class="kw">take </span><font color="Maroon" title="c21">j</font> = <font color="Maroon" title="c19">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font>) )</span><br/><span class="kw">thus </span><a NAME="E5:5_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_3"><span class="lab"><font color="Green" title="E36">A55</font></span></a>, <a class="txt" href="#E4:5_3"><span class="lab"><font color="Green" title="E37">A56</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">g1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E42:5"/><span class="lab"><font color="Green" title="E36">A57</font></span>: 
( <font color="Maroon" title="c17">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">f1</font> &amp; <font color="Maroon" title="c17">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c17">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E43:5"/><span class="lab"><font color="Green" title="E37">A58</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</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="c17">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E5:5"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="goboard2.html#T12" title="GOBOARD2:th.12">GOBOARD2:12</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">n</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">f2</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) ) )</span><br/><span class="kw">assume </span><a NAME="E1:5_4"/><span class="lab"><font color="Green" title="E38">A59</font></span>: 
<font color="Maroon" title="c18">n</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">f2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )</span><br/><a NAME="E2:5_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">n</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="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">i</font>, <font color="Maroon" title="c20">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:5_4"/><span class="lab"><font color="Green" title="E39">A60</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i</font>,<font color="Maroon" title="c20">j</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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c19">i</font>,<font color="Maroon" title="c20">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T14" title="GOBOARD2:th.14">GOBOARD2:14</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c21">i</font> = <font color="Maroon" title="c19">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i</font>,<font color="Olive" title="b1">j</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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c21">i</font>,<font color="Olive" title="b1">j</font>) )</span><br/><span class="kw">take </span><font color="Maroon" title="c22">j</font> = <font color="Maroon" title="c20">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i</font>,<font color="Maroon" title="c22">j</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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c21">i</font>,<font color="Maroon" title="c22">j</font>) )</span><br/><span class="kw">thus </span><a NAME="E5:5_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i</font>,<font color="Maroon" title="c22">j</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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c21">i</font>,<font color="Maroon" title="c22">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_4"><span class="lab"><font color="Green" title="E38">A59</font></span></a>, <a class="txt" href="#E4:5_4"><span class="lab"><font color="Green" title="E39">A60</font></span></a>, <a class="ref" href="finseq_4.html#T69" title="FINSEQ_4:th.69">FINSEQ_4:69</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">g2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E46:5"/><span class="lab"><font color="Green" title="E38">A61</font></span>: 
( <font color="Maroon" title="c18">g2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f2</font> &amp; <font color="Maroon" title="c18">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c18">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E47:5"/><span class="lab"><font color="Green" title="E39">A62</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</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="c18">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E6:5"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="goboard2.html#T12" title="GOBOARD2:th.12">GOBOARD2:12</a></span>;<br/>
<a NAME="E48:5"/><span class="lab"><font color="Green" title="E40">A63</font></span>: 
2 <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="c18">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E47:5"><span class="lab"><font color="Green" title="E39">A62</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E49:5"/><span class="lab"><font color="Green" title="E41">A64</font></span>: 
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">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E50:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A65</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E51:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A66</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <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="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T69" title="FINSEQ_4:th.69">FINSEQ_4:69</a></span>;<br/>
<a NAME="E52:5"/><span class="lab"><font color="Green" title="E44">A67</font></span>: 
(  <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> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/>
<a NAME="E53:5"/><span class="lab"><font color="Green" title="E45">A68</font></span>: 
(  <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> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</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>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/>
<a NAME="E54:5"/><span class="lab"><font color="Green" title="E46">A69</font></span>: 
(  <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> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c4">f2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c4">f2</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</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>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/>
<a NAME="E55:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A70</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E50:5"><span class="lab"><font color="Green" title="E42">A65</font></span></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E48">A71</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">m</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="c5">f</font> implies <font color="Maroon" title="c8">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c20">s</font> = <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font>;<br/><span class="kw">assume </span><a NAME="E1:5_5"/><span class="lab"><font color="Green" title="E49">A72</font></span>: 
<font color="Maroon" title="c19">m</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="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font></span><br/><a NAME="E2:5_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A73</font></span>: 
<font color="Maroon" title="c19">m</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">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:5_5"/><span class="lab"><font color="Green" title="E51">A74</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_5"><span class="lab"><font color="Green" title="E49">A72</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_5_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="E1:5_5_1_1"/>
( <font color="Maroon" title="c19">m</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="c3">f1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:5_5_1_1_1"/><span class="lab"><font color="Green" title="E52">A75</font></span>: 
<font color="Maroon" title="c19">m</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="c3">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c21">u</font> = <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">m</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> ;<br/><a NAME="E3:5_5_1_1_1"/><span class="lab"><font color="Green" title="E53">A76</font></span>: 
<font color="Maroon" title="c19">m</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="c3">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_5"><span class="lab"><font color="Green" title="E51">A74</font></span></a>, <a class="txt" href="#E1:5_5_1_1_1"><span class="lab"><font color="Green" title="E52">A75</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:5_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E5:5_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A77</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:5"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E52:5"><span class="lab"><font color="Green" title="E44">A67</font></span></a>, <a class="txt" href="#E1:5_5"><span class="lab"><font color="Green" title="E49">A72</font></span></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/><a NAME="E6:5_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:5"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E53:5"><span class="lab"><font color="Green" title="E45">A68</font></span></a>, <a class="txt" href="#E3:5_5_1_1_1"><span class="lab"><font color="Green" title="E53">A76</font></span></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:5_5_1_1_1"/>
<font color="Maroon" title="c8">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:5"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E16:5"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E53:5"><span class="lab"><font color="Green" title="E45">A68</font></span></a>, <a class="txt" href="#E55:5"><span class="lab"><font color="Green" title="E47">A70</font></span></a>, <a class="txt" href="#E3:5_5_1_1_1"><span class="lab"><font color="Green" title="E53">A76</font></span></a>, <a class="txt" href="#E5:5_5_1_1_1"><span class="lab"><font color="Green" title="E54">A77</font></span></a>, <a class="ref" href="goboard4.html#D2" target="_self" title="GOBOARD4:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="5_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:5_5_1_1_2"/><span class="lab"><font color="Green" title="E52">A78</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">w5</font> = <font color="Maroon" title="c19">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/><a NAME="E3:5_5_1_1_2"/>
<font color="Maroon" title="c21">w5</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_5_1_1_2"><span class="lab"><font color="Green" title="E52">A78</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E4:5_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A79</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">w5</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E5:5_5_1_1_2"/><span class="lab"><font color="Green" title="E54">A80</font></span>: 
<font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">w5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">m1</font> = <font color="Maroon" title="c19">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">u</font> = <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">m1</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> ;<br/><a NAME="E8:5_5_1_1_2"/><span class="lab"><font color="Green" title="E55">A81</font></span>: 
<font color="Maroon" title="c21">w5</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">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:5"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="txt" href="#E2:5_5"><span class="lab"><font color="Green" title="E50">A73</font></span></a>, <a class="txt" href="#E5:5_5_1_1_2"><span class="lab"><font color="Green" title="E54">A80</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E9:5_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">w5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:5_5_1_1_2"><span class="lab"><font color="Green" title="E54">A80</font></span></a>, <a class="txt" href="#E4:5_5_1_1_2"><span class="lab"><font color="Green" title="E53">A79</font></span></a>, <a class="ref" href="seq_4.html#T136" title="SEQ_4:th.136">SEQ_4:136</a></span>;<br/><a NAME="E10:5_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A82</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:5"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E52:5"><span class="lab"><font color="Green" title="E44">A67</font></span></a>, <a class="txt" href="#E1:5_5"><span class="lab"><font color="Green" title="E49">A72</font></span></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/><a NAME="E11:5_5_1_1_2"/><span class="lab"><font color="Green" title="E57">A83</font></span>: 
<font color="Maroon" title="c22">m1</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">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_5_1_1_2"><span class="lab"><font color="Green" title="E53">A79</font></span></a>, <a class="txt" href="#E8:5_5_1_1_2"><span class="lab"><font color="Green" title="E55">A81</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E12:5_5_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E54:5"><span class="lab"><font color="Green" title="E46">A69</font></span></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E13:5_5_1_1_2"/>
<font color="Maroon" title="c8">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:5"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E54:5"><span class="lab"><font color="Green" title="E46">A69</font></span></a>, <a class="txt" href="#E55:5"><span class="lab"><font color="Green" title="E47">A70</font></span></a>, <a class="txt" href="#E11:5_5_1_1_2"><span class="lab"><font color="Green" title="E57">A83</font></span></a>, <a class="txt" href="#E10:5_5_1_1_2"><span class="lab"><font color="Green" title="E56">A82</font></span></a>, <a class="ref" href="goboard4.html#D2" target="_self" title="GOBOARD4:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E5:5_5"/>
<font color="Maroon" title="c8">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E57:5"/><span class="lab"><font color="Green" title="E49">A84</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <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="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:5"><span class="lab"><font color="Green" title="E42">A65</font></span></a>, <a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></span>;<br/>
<a NAME="E58:5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <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="c8">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:5"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E52:5"><span class="lab"><font color="Green" title="E44">A67</font></span></a>, <a class="txt" href="#E51:5"><span class="lab"><font color="Green" title="E43">A66</font></span></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/>
<a NAME="E59:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A85</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:5"><span class="lab"><font color="Green" title="E49">A84</font></span></a>, <a class="txt" href="#E56:5"><span class="lab"><font color="Green" title="E48">A71</font></span></a>, <a class="ref" href="goboard2.html#T17" title="GOBOARD2:th.17">GOBOARD2:17</a></span>;<br/>
<a NAME="E60:5"/><span class="lab"><font color="Green" title="E51">A86</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</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">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:5"><span class="lab"><font color="Green" title="E41">A64</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E61:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A87</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T69" title="FINSEQ_4:th.69">FINSEQ_4:69</a></span>;<br/>
<a NAME="E62:5"/><span class="lab"><font color="Green" title="E53">A88</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E54:5"><span class="lab"><font color="Green" title="E46">A69</font></span></a>, <a class="txt" href="#E60:5"><span class="lab"><font color="Green" title="E51">A86</font></span></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E54">A89</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">m</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="c5">f</font> implies <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><span class="kw">set </span><font color="Maroon" title="c20">s</font> = <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font>;<br/><span class="kw">assume </span><a NAME="E1:5_6"/><span class="lab"><font color="Green" title="E55">A90</font></span>: 
<font color="Maroon" title="c19">m</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="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><a NAME="E2:5_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A91</font></span>: 
<font color="Maroon" title="c19">m</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">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:5_6"/><span class="lab"><font color="Green" title="E57">A92</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_6"><span class="lab"><font color="Green" title="E55">A90</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_6_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_6_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="E1:5_6_1_1"/>
( <font color="Maroon" title="c19">m</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="c3">f1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_6_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:5_6_1_1_1"/><span class="lab"><font color="Green" title="E58">A93</font></span>: 
<font color="Maroon" title="c19">m</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="c3">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c21">u</font> = <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">m</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> ;<br/><a NAME="E3:5_6_1_1_1"/><span class="lab"><font color="Green" title="E59">A94</font></span>: 
<font color="Maroon" title="c19">m</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="c3">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_6"><span class="lab"><font color="Green" title="E57">A92</font></span></a>, <a class="txt" href="#E1:5_6_1_1_1"><span class="lab"><font color="Green" title="E58">A93</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:5_6_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E5:5_6_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A95</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:5"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E52:5"><span class="lab"><font color="Green" title="E44">A67</font></span></a>, <a class="txt" href="#E1:5_6"><span class="lab"><font color="Green" title="E55">A90</font></span></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/><a NAME="E6:5_6_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:5"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E53:5"><span class="lab"><font color="Green" title="E45">A68</font></span></a>, <a class="txt" href="#E3:5_6_1_1_1"><span class="lab"><font color="Green" title="E59">A94</font></span></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:5_6_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:5"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E16:5"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E53:5"><span class="lab"><font color="Green" title="E45">A68</font></span></a>, <a class="txt" href="#E62:5"><span class="lab"><font color="Green" title="E53">A88</font></span></a>, <a class="txt" href="#E3:5_6_1_1_1"><span class="lab"><font color="Green" title="E59">A94</font></span></a>, <a class="txt" href="#E5:5_6_1_1_1"><span class="lab"><font color="Green" title="E60">A95</font></span></a>, <a class="ref" href="goboard4.html#D2" target="_self" title="GOBOARD4:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="5_6_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:5_6_1_1_2"/><span class="lab"><font color="Green" title="E58">A96</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">w5</font> = <font color="Maroon" title="c19">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/><a NAME="E3:5_6_1_1_2"/>
<font color="Maroon" title="c21">w5</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_6_1_1_2"><span class="lab"><font color="Green" title="E58">A96</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E4:5_6_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A97</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">w5</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E5:5_6_1_1_2"/><span class="lab"><font color="Green" title="E60">A98</font></span>: 
<font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">w5</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">m1</font> = <font color="Maroon" title="c19">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">u</font> = <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">m1</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> ;<br/><a NAME="E8:5_6_1_1_2"/><span class="lab"><font color="Green" title="E61">A99</font></span>: 
<font color="Maroon" title="c21">w5</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">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:5"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="txt" href="#E2:5_6"><span class="lab"><font color="Green" title="E56">A91</font></span></a>, <a class="txt" href="#E5:5_6_1_1_2"><span class="lab"><font color="Green" title="E60">A98</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E9:5_6_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">w5</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:5_6_1_1_2"><span class="lab"><font color="Green" title="E60">A98</font></span></a>, <a class="txt" href="#E4:5_6_1_1_2"><span class="lab"><font color="Green" title="E59">A97</font></span></a>, <a class="ref" href="seq_4.html#T136" title="SEQ_4:th.136">SEQ_4:136</a></span>;<br/><a NAME="E10:5_6_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A100</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:5"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E52:5"><span class="lab"><font color="Green" title="E44">A67</font></span></a>, <a class="txt" href="#E1:5_6"><span class="lab"><font color="Green" title="E55">A90</font></span></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/><a NAME="E11:5_6_1_1_2"/><span class="lab"><font color="Green" title="E63">A101</font></span>: 
<font color="Maroon" title="c22">m1</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">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5_6_1_1_2"><span class="lab"><font color="Green" title="E59">A97</font></span></a>, <a class="txt" href="#E8:5_6_1_1_2"><span class="lab"><font color="Green" title="E61">A99</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E12:5_6_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E54:5"><span class="lab"><font color="Green" title="E46">A69</font></span></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E13:5_6_1_1_2"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:5"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E54:5"><span class="lab"><font color="Green" title="E46">A69</font></span></a>, <a class="txt" href="#E62:5"><span class="lab"><font color="Green" title="E53">A88</font></span></a>, <a class="txt" href="#E11:5_6_1_1_2"><span class="lab"><font color="Green" title="E63">A101</font></span></a>, <a class="txt" href="#E10:5_6_1_1_2"><span class="lab"><font color="Green" title="E62">A100</font></span></a>, <a class="ref" href="goboard4.html#D2" target="_self" title="GOBOARD4:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E5:5_6"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E64:5"/><span class="lab"><font color="Green" title="E55">A102</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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">f2</font></span>)</span> <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="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:5"><span class="lab"><font color="Green" title="E51">A86</font></span></a>, <a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></span>;<br/>
<a NAME="E65:5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:5"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E52:5"><span class="lab"><font color="Green" title="E44">A67</font></span></a>, <a class="txt" href="#E61:5"><span class="lab"><font color="Green" title="E52">A87</font></span></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/>
<a NAME="E66:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A103</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> (<span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:5"><span class="lab"><font color="Green" title="E55">A102</font></span></a>, <a class="txt" href="#E63:5"><span class="lab"><font color="Green" title="E54">A89</font></span></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></span>;<br/>
<a NAME="E67:5"/>
2 <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="c17">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E43:5"><span class="lab"><font color="Green" title="E37">A58</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E68:5"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f1</font> <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="c2">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:5"><span class="lab"><font color="Green" title="E36">A57</font></span></a>, <a class="txt" href="#E46:5"><span class="lab"><font color="Green" title="E38">A61</font></span></a>, <a class="txt" href="#E19:5"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="txt" href="#E34:5"><span class="lab"><font color="Green" title="E31">A38</font></span></a>, <a class="txt" href="#E51:5"><span class="lab"><font color="Green" title="E43">A66</font></span></a>, <a class="txt" href="#E61:5"><span class="lab"><font color="Green" title="E52">A87</font></span></a>, <a class="txt" href="#E32:5"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="txt" href="#E39:5"><span class="lab"><font color="Green" title="E35">A54</font></span></a>, <a class="txt" href="#E59:5"><span class="lab"><font color="Green" title="E50">A85</font></span></a>, <a class="txt" href="#E66:5"><span class="lab"><font color="Green" title="E56">A103</font></span></a>, <a class="txt" href="#E48:5"><span class="lab"><font color="Green" title="E40">A63</font></span></a>, <a class="ref" href="goboard4.html#T2" target="_self" title="GOBOARD4:th.2">Th2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
