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<span class="kw">let </span><font color="Maroon" title="c1">P1</font>, <font color="Maroon" title="c2">P2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">q1</font>, <font color="Olive" title="b4">q2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c1">P1</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b3">q1</font> &amp; <font color="Maroon" title="c2">P2</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Olive" title="b2">p2</font>,<font color="Olive" title="b4">q2</font> &amp; ( for <font color="Olive" title="b5">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">P2</font> holds <br/>( <font color="Olive" title="b1">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> <font color="Olive" title="b5">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b5">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) ) &amp; ( for <font color="Olive" title="b5">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">P2</font> holds <br/>( <font color="Olive" title="b2">p2</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> <font color="Olive" title="b5">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b5">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) ) holds <br/><font color="Maroon" title="c1">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">P2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font>, <font color="Maroon" title="c5">q1</font>, <font color="Maroon" title="c6">q2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">P1</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font> &amp; <font color="Maroon" title="c2">P2</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font> &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">P2</font> holds <br/>( <font color="Maroon" title="c3">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> <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) ) &amp; ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">P2</font> holds <br/>( <font color="Maroon" title="c4">p2</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> <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) ) implies <font color="Maroon" title="c1">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">P2</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:9"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c1">P1</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font>
 <span class="kw">and </span><br/><a NAME="E2:9"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c2">P2</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>
 <span class="kw">and </span><br/><a NAME="E3:9"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">P2</font> holds <br/>( <font color="Maroon" title="c3">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> <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E4:9"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">P2</font> holds <br/>( <font color="Maroon" title="c4">p2</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> <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">P2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">f1</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="E6:9"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c7">f1</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a> 
 <span class="kw">and </span><br/><a NAME="E7:9"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
<font color="Maroon" title="c1">P1</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="c7">f1</font>
 <span class="kw">and </span><br/><a NAME="E8:9"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">and </span><br/><a NAME="E9:9"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
<font color="Maroon" title="c5">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="c7">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="topreal4.html#D1" title="TOPREAL4:def.1">TOPREAL4:def 1</a></span>;<br/>
<a NAME="E10:9"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c7">f1</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="E12:9"/><span class="lab"><font color="Green" title="E15">A9</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="c8">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="c8">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/>
<span class="kw">consider </span><font color="Maroon" title="c9">f2</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="E14:9"/><span class="lab"><font color="Green" title="E16">A10</font></span>: 
<font color="Maroon" title="c9">f2</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a> 
 <span class="kw">and </span><br/><a NAME="E15:9"/><span class="lab"><font color="Green" title="E17">A11</font></span>: 
<font color="Maroon" title="c2">P2</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="c9">f2</font>
 <span class="kw">and </span><br/><a NAME="E16:9"/><span class="lab"><font color="Green" title="E18">A12</font></span>: 
<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">and </span><br/><a NAME="E17:9"/><span class="lab"><font color="Green" title="E19">A13</font></span>: 
<font color="Maroon" title="c6">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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="c9">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="topreal4.html#D1" title="TOPREAL4:def.1">TOPREAL4:def 1</a></span>;<br/>
<a NAME="E18:9"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">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="#E14:9"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f2</font> = <font color="Maroon" title="c9">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="E20:9"/><span class="lab"><font color="Green" title="E20">A14</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="c10">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="c10">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">set </span><font color="Maroon" title="c11">x1</font> =  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">f1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">y1</font> =  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">f1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">x2</font> =  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">f2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">y2</font> =  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</font>;<br/>
<a NAME="E21:9"/><span class="lab"><font color="Green" title="E21">A15</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="c8">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="c8">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="c8">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="c8">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#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></span>;<br/>
<a NAME="E22:9"/><span class="lab"><font color="Green" title="E22">A16</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#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</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#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</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#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">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="c10">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#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></span>;<br/>
<a NAME="E23:9"/><span class="lab"><font color="Green" title="E23">A17</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="c10">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:9"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></span>;<br/>
<a NAME="E24:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A18</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="c10">f2</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="E25:9"/><span class="kw">then </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="c10">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E26:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A19</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">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="c4">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:9"><span class="lab"><font color="Green" title="E18">A12</font></span></a>, <a class="txt" href="#E20:9"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="txt" href="#E22:9"><span class="lab"><font color="Green" title="E22">A16</font></span></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></span>;<br/>
<a NAME="E27:9"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">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="c10">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:9"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E28:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A20</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">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="c10">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="#E17:9"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="txt" href="#E20:9"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="txt" href="#E22:9"><span class="lab"><font color="Green" title="E22">A16</font></span></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></span>;<br/>
<a NAME="E29:9"/><span class="lab"><font color="Green" title="E27">A21</font></span>: 
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">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="c10">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="c10">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="c10">f2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="goboard4.html#D2" target="_self" title="GOBOARD4:def.2">GOBOARD4:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> implies ( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <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="c10">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="c10">f2</font></span>)</span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c16">q</font> = <font color="Maroon" title="c10">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font>;<br/>
<span class="kw">assume </span><a NAME="E1:9_1"/><span class="lab"><font color="Green" title="E28">A22</font></span>: 
<font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <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="c10">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="c10">f2</font></span>)</span> )</span><br/>

<a NAME="E2:9_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c10">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:9"><span class="lab"><font color="Green" title="E23">A17</font></span></a>, <a class="txt" href="#E20:9"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="txt" href="#E22:9"><span class="lab"><font color="Green" title="E22">A16</font></span></a>, <a class="ref" href="goboard1.html#T16" title="GOBOARD1:th.16">GOBOARD1:16</a></span>;<br/>
<a NAME="E3:9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A23</font></span>: 
<font color="Maroon" title="c10">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:9"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E4:9_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1"><span class="lab"><font color="Green" title="E28">A22</font></span></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:9_1"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <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="c10">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="c10">f2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E26:9"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="txt" href="#E28:9"><span class="lab"><font color="Green" title="E26">A20</font></span></a>, <a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E29">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E30:9"/><span class="lab"><font color="Green" title="E28">A24</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="c8">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></span>;<br/>
<a NAME="E31:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A25</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="c8">f1</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="E32:9"/><span class="kw">then </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="c8">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E33:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A26</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">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="c3">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="#E8:9"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="txt" href="#E12:9"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="txt" href="#E21:9"><span class="lab"><font color="Green" title="E21">A15</font></span></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></span>;<br/>
<a NAME="E34:9"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c8">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:9"><span class="lab"><font color="Green" title="E29">A25</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E35:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A27</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">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="c8">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:9"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="txt" href="#E12:9"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="txt" href="#E21:9"><span class="lab"><font color="Green" title="E21">A15</font></span></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></span>;<br/>
<a NAME="E36:9"/><span class="lab"><font color="Green" title="E32">A28</font></span>: 
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">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="c8">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="c8">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="c8">f1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="goboard4.html#D2" target="_self" title="GOBOARD4:def.2">GOBOARD4:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> implies ( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <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="c8">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="c8">f1</font></span>)</span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c16">q</font> = <font color="Maroon" title="c8">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font>;<br/>
<span class="kw">assume </span><a NAME="E1:9_2"/><span class="lab"><font color="Green" title="E33">A29</font></span>: 
<font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <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="c8">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="c8">f1</font></span>)</span> )</span><br/>

<a NAME="E2:9_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c8">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:9"><span class="lab"><font color="Green" title="E28">A24</font></span></a>, <a class="txt" href="#E12:9"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="txt" href="#E21:9"><span class="lab"><font color="Green" title="E21">A15</font></span></a>, <a class="ref" href="goboard1.html#T16" title="GOBOARD1:th.16">GOBOARD1:16</a></span>;<br/>
<a NAME="E3:9_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A30</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E4:9_2"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_2"><span class="lab"><font color="Green" title="E33">A29</font></span></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:9_2"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <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="c8">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="c8">f1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E33:9"><span class="lab"><font color="Green" title="E30">A26</font></span></a>, <a class="txt" href="#E35:9"><span class="lab"><font color="Green" title="E31">A27</font></span></a>, <a class="txt" href="#E3:9_2"><span class="lab"><font color="Green" title="E34">A30</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E37:9"/><span class="lab"><font color="Green" title="E33">A31</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="c10">f2</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="c10">f2</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="c10">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="c10">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#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></span>;<br/>
<a NAME="E38:9"/><span class="lab"><font color="Green" title="E34">A32</font></span>: 
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">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="c8">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="c8">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="c8">f1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="goboard4.html#D2" target="_self" title="GOBOARD4:def.2">GOBOARD4:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> implies ( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <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="c8">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="c8">f1</font></span>)</span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c16">q</font> = <font color="Maroon" title="c10">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font>;<br/>
<span class="kw">assume </span><a NAME="E1:9_3"/><span class="lab"><font color="Green" title="E35">A33</font></span>: 
<font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <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="c8">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="c8">f1</font></span>)</span> )</span><br/>

<a NAME="E2:9_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c10">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:9"><span class="lab"><font color="Green" title="E23">A17</font></span></a>, <a class="txt" href="#E20:9"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="txt" href="#E37:9"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="ref" href="goboard1.html#T16" title="GOBOARD1:th.16">GOBOARD1:16</a></span>;<br/>
<a NAME="E3:9_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A34</font></span>: 
<font color="Maroon" title="c10">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:9"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E4:9_3"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_3"><span class="lab"><font color="Green" title="E35">A33</font></span></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:9_3"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <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="c8">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="c8">f1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E33:9"><span class="lab"><font color="Green" title="E30">A26</font></span></a>, <a class="txt" href="#E35:9"><span class="lab"><font color="Green" title="E31">A27</font></span></a>, <a class="txt" href="#E3:9_3"><span class="lab"><font color="Green" title="E36">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E39:9"/><span class="lab"><font color="Green" title="E35">A35</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#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">f1</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#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">f1</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#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">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="c8">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#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></span>;<br/>
<a NAME="E40:9"/><span class="lab"><font color="Green" title="E36">A36</font></span>: 
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">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="c10">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="c10">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="c10">f2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="goboard4.html#D2" target="_self" title="GOBOARD4:def.2">GOBOARD4:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> implies ( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <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="c10">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="c10">f2</font></span>)</span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c16">q</font> = <font color="Maroon" title="c8">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font>;<br/>
<span class="kw">assume </span><a NAME="E1:9_4"/><span class="lab"><font color="Green" title="E37">A37</font></span>: 
<font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <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="c10">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="c10">f2</font></span>)</span> )</span><br/>

<a NAME="E2:9_4"/><span class="kw">then </span>
<font color="Maroon" title="c8">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c8">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:9"><span class="lab"><font color="Green" title="E28">A24</font></span></a>, <a class="txt" href="#E12:9"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="txt" href="#E39:9"><span class="lab"><font color="Green" title="E35">A35</font></span></a>, <a class="ref" href="goboard1.html#T16" title="GOBOARD1:th.16">GOBOARD1:16</a></span>;<br/>
<a NAME="E3:9_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A38</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E4:9_4"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_4"><span class="lab"><font color="Green" title="E37">A37</font></span></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:9_4"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <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="c10">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="c10">f2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E26:9"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="txt" href="#E28:9"><span class="lab"><font color="Green" title="E26">A20</font></span></a>, <a class="txt" href="#E3:9_4"><span class="lab"><font color="Green" title="E38">A38</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E41:9"/><span class="lab"><font color="Green" title="E37">A39</font></span>: 
( <font color="Maroon" title="c10">f2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c10">f2</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:9"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></span>;<br/>
<a NAME="E42:9"/>
( <font color="Maroon" title="c8">f1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c8">f1</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E43:9"/>
<font color="Maroon" title="c1">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E15:9"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="txt" href="#E41:9"><span class="lab"><font color="Green" title="E37">A39</font></span></a>, <a class="txt" href="#E30:9"><span class="lab"><font color="Green" title="E28">A24</font></span></a>, <a class="txt" href="#E23:9"><span class="lab"><font color="Green" title="E23">A17</font></span></a>, <a class="txt" href="#E36:9"><span class="lab"><font color="Green" title="E32">A28</font></span></a>, <a class="txt" href="#E38:9"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="txt" href="#E40:9"><span class="lab"><font color="Green" title="E36">A36</font></span></a>, <a class="txt" href="#E29:9"><span class="lab"><font color="Green" title="E27">A21</font></span></a>, <a class="ref" href="goboard4.html#T5" target="_self" title="GOBOARD4:th.5">Th5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
