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<span class="kw">let </span><font color="Maroon" title="c1">P</font>, <font color="Maroon" title="c2">Q</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <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">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c2">Q</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b3">q1</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">P</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="b2">p2</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="c2">Q</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="b2">p2</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">P</font> holds <br/>( <font color="Olive" title="b3">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> <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>  ) ) &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="c2">Q</font> holds <br/>( <font color="Olive" title="b3">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> <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">P</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c2">Q</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">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c2">Q</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c5">q1</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">P</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="c4">p2</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="c2">Q</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="c4">p2</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">P</font> holds <br/>( <font color="Maroon" title="c5">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> <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>  ) ) &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="c2">Q</font> holds <br/>( <font color="Maroon" title="c5">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> <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">P</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c2">Q</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:67"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:67"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c2">Q</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>
 <span class="kw">and </span><br/><a NAME="E3:67"/><span class="lab"><font color="Green" title="E45">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">P</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="c4">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E4:67"/><span class="lab"><font color="Green" title="E46">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="c2">Q</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="c4">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E5:67"/><span class="lab"><font color="Green" title="E47">A5</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">P</font> holds <br/>( <font color="Maroon" title="c5">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> <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>  )
 <span class="kw">and </span><br/><a NAME="E6:67"/><span class="lab"><font color="Green" title="E48">A6</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="c2">Q</font> holds <br/>( <font color="Maroon" title="c5">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> <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">P</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c2">Q</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:67"/><span class="lab"><font color="Green" title="E49">A7</font></span>: 
<font color="Maroon" title="c7">g</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E9:67"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q1</font>
 <span class="kw">and </span><br/><a NAME="E10:67"/><span class="lab"><font color="Green" title="E51">A9</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></span>;<br/>
<a NAME="E11:67"/><span class="lab"><font color="Green" title="E52">A10</font></span>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D8" title="EUCLID:def.8">EUCLID:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">P9</font> = <font color="Maroon" title="c1">P</font>, <font color="Maroon" title="c9">Q9</font> = <font color="Maroon" title="c2">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span></span>)</span> ;<br/>
<a NAME="E13:67"/>
<font color="Maroon" title="c1">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="jordan5a.html#T1" title="JORDAN5A:th.1">JORDAN5A:1</a></span>;<br/>
<a NAME="E14:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A11</font></span>: 
<font color="Maroon" title="c8">P9</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:67"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="compts_1.html#T33" title="COMPTS_1:th.33">COMPTS_1:33</a></span>;<br/>
<a NAME="E15:67"/>
<font color="Maroon" title="c2">Q</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="jordan5a.html#T1" title="JORDAN5A:th.1">JORDAN5A:1</a></span>;<br/>
<a NAME="E16:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A12</font></span>: 
<font color="Maroon" title="c9">Q9</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:67"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="compts_1.html#T33" title="COMPTS_1:th.33">COMPTS_1:33</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">e</font> = <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E18:67"/><span class="lab"><font color="Green" title="E55">A13</font></span>: 
<font color="Maroon" title="c11">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E19:67"/><span class="lab"><font color="Green" title="E56">A14</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font>
 <span class="kw">and </span><br/><a NAME="E20:67"/><span class="lab"><font color="Green" title="E57">A15</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">f1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<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="E22:67"/><span class="lab"><font color="Green" title="E58">A16</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c12">f1</font>
 <span class="kw">and </span><br/><a NAME="E23:67"/><span class="lab"><font color="Green" title="E59">A17</font></span>: 
<font color="Maroon" title="c12">f1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E24:67"/>
<font color="Maroon" title="c12">f1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="ref" href="jordan7.html#T15" title="JORDAN7:th.15">JORDAN7:15</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">g1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<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="E26:67"/><span class="lab"><font color="Green" title="E60">A18</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c13">g1</font>
 <span class="kw">and </span><br/><a NAME="E27:67"/><span class="lab"><font color="Green" title="E61">A19</font></span>: 
<font color="Maroon" title="c13">g1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E28:67"/>
<font color="Maroon" title="c13">g1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="jordan7.html#T15" title="JORDAN7:th.15">JORDAN7:15</a></span>;<br/>
<span class="kw">assume </span><a NAME="E29:67"/>
<font color="Maroon" title="c1">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E30:67"/><span class="kw">then </span>
<font color="Maroon" title="c1">P</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c2">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E31:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A20</font></span>: 
 <a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</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="#E14:67"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E16:67"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="ref" href="jgraph_1.html#T55" target="_self" title="JGRAPH_1:th.55">Th55</a></span>;<br/>
<a NAME="E32:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A21</font></span>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">hb</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E34:67"/><span class="lab"><font color="Green" title="E64">A22</font></span>: 
<font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E35:67"/><span class="lab"><font color="Green" title="E65">A23</font></span>: 
<font color="Maroon" title="c14">hb</font> <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="c14">hb</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E36:67"/><span class="lab"><font color="Green" title="E66">A24</font></span>: 
5 <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="c14">hb</font>
 <span class="kw">and </span><br/><a NAME="E37:67"/><span class="lab"><font color="Green" title="E67">A25</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">hb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 <span class="kw">and </span><br/><a NAME="E38:67"/><span class="lab"><font color="Green" title="E68">A26</font></span>: 
<font color="Maroon" title="c14">hb</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 <span class="kw">and </span><br/><a NAME="E39:67"/><span class="lab"><font color="Green" title="E69">A27</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">R</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <br/> for <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">hb</font> &amp; <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">R</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b3">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:67"><span class="lab"><font color="Green" title="E61">A19</font></span></a>, <a class="ref" href="uniform1.html#T15" title="UNIFORM1:th.15">UNIFORM1:15</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">h</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E41:67"/><span class="lab"><font color="Green" title="E70">A28</font></span>: 
<font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E42:67"/><span class="lab"><font color="Green" title="E71">A29</font></span>: 
<font color="Maroon" title="c15">h</font> <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="c15">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E43:67"/><span class="lab"><font color="Green" title="E72">A30</font></span>: 
5 <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="c15">h</font>
 <span class="kw">and </span><br/><a NAME="E44:67"/><span class="lab"><font color="Green" title="E73">A31</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c15">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 <span class="kw">and </span><br/><a NAME="E45:67"/><span class="lab"><font color="Green" title="E74">A32</font></span>: 
<font color="Maroon" title="c15">h</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 <span class="kw">and </span><br/><a NAME="E46:67"/><span class="lab"><font color="Green" title="E75">A33</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">R</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <br/> for <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">h</font> &amp; <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">R</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b3">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:67"><span class="lab"><font color="Green" title="E59">A17</font></span></a>, <a class="txt" href="#E32:67"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="ref" href="uniform1.html#T15" title="UNIFORM1:th.15">UNIFORM1:15</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1</span>)</span>;<br/>
<a NAME="E47:67"/>
 ex <font color="Olive" title="b1">h19</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h19</font> <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="c15">h</font> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">i</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="Olive" title="b1">h19</font> holds <br/><font color="Olive" title="b1">h19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">i</font>) ) )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">h19</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E49:67"/><span class="lab"><font color="Green" title="E76">A34</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">h19</font> <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="c15">h</font>
 <span class="kw">and </span><br/><a NAME="E50:67"/><span class="lab"><font color="Green" title="E77">A35</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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="c16">h19</font> holds <br/><font color="Maroon" title="c16">h19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 ;<br/>
<span class="lab"><font color="Green" title="E78">A36</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">g1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 

;<br/>
<a NAME="E52:67"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">h19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">h19</font> or <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:67_2"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">h19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:67_2"/><span class="lab"><font color="Green" title="E79">A37</font></span>: 
<font color="Maroon" title="c18">x</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="c16">h19</font>
 <span class="kw">and </span><br/><a NAME="E4:67_2"/><span class="lab"><font color="Green" title="E80">A38</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">h19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">i</font> = <font color="Maroon" title="c18">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_2"><span class="lab"><font color="Green" title="E79">A37</font></span></a></span>;<br/>
<a NAME="E6:67_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">h19</font> <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> <font color="Maroon" title="c16">h19</font></span>)</span>
 
<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="E7:67_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">i</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="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E3:67_2"><span class="lab"><font color="Green" title="E79">A37</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:67_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A39</font></span>: 
<font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="lab"><font color="Green" title="E82">A40</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">f1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 

;<br/>
<span class="lab"><font color="Green" title="E83">A41</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c1">P</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>
;<br/>
<a NAME="E11:67_2"/>
<font color="Maroon" title="c16">h19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:67"><span class="lab"><font color="Green" title="E77">A35</font></span></a>, <a class="txt" href="#E3:67_2"><span class="lab"><font color="Green" title="E79">A37</font></span></a></span>;<br/>
<a NAME="E12:67_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">h19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E44:67"><span class="lab"><font color="Green" title="E73">A31</font></span></a>, <a class="txt" href="#E8:67_2"><span class="lab"><font color="Green" title="E81">A39</font></span></a>, <a class="txt" href="#E9:67_2"><span class="lab"><font color="Green" title="E82">A40</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:67_2"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_2"><span class="lab"><font color="Green" title="E80">A38</font></span></a>, <a class="txt" href="#E10:67_2"><span class="lab"><font color="Green" title="E83">A41</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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">h1</font> = <font color="Maroon" title="c16">h19</font> as   <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">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E54:67"/><span class="lab"><font color="Green" title="E79">A42</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:67"><span class="lab"><font color="Green" title="E72">A30</font></span></a>, <a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E55:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A43</font></span>: 
<font color="Maroon" title="c17">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">h1</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#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E56:67"/><span class="lab"><font color="Green" title="E81">A44</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c18">Pa</font> = <font color="Maroon" title="c1">P</font> as   <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> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">W1</font> = <font color="Maroon" title="c1">P</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c20">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font> implies <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5 )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:67_3"/><span class="lab"><font color="Green" title="E82">A45</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i</font>
 <span class="kw">and </span><br/><a NAME="E4:67_3"/><span class="lab"><font color="Green" title="E83">A46</font></span>: 
<font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span><br/>

<a NAME="E5:67_3"/><span class="lab"><font color="Green" title="E84">A47</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_3"><span class="lab"><font color="Green" title="E82">A45</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E6:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A48</font></span>: 
<font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E4:67_3"><span class="lab"><font color="Green" title="E83">A46</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E7:67_3"/><span class="lab"><font color="Green" title="E86">A49</font></span>: 
<font color="Maroon" title="c20">i</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="c16">h19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_3"><span class="lab"><font color="Green" title="E83">A46</font></span></a>, <a class="txt" href="#E5:67_3"><span class="lab"><font color="Green" title="E84">A47</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E8:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A50</font></span>: 
<font color="Maroon" title="c16">h19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:67"><span class="lab"><font color="Green" title="E77">A35</font></span></a></span>;<br/>
<a NAME="E9:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A51</font></span>: 
<font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_3"><span class="lab"><font color="Green" title="E83">A46</font></span></a>, <a class="txt" href="#E5:67_3"><span class="lab"><font color="Green" title="E84">A47</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E10:67_3"/><span class="lab"><font color="Green" title="E89">A52</font></span>: 
<font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_3"><span class="lab"><font color="Green" title="E83">A46</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E11:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A53</font></span>: 
<font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E3:67_3"><span class="lab"><font color="Green" title="E82">A45</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E12:67_3"/><span class="lab"><font color="Green" title="E91">A54</font></span>: 
<font color="Maroon" title="c20">i</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="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E3:67_3"><span class="lab"><font color="Green" title="E82">A45</font></span></a>, <a class="txt" href="#E10:67_3"><span class="lab"><font color="Green" title="E89">A52</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E13:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A55</font></span>: 
<font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E14:67_3"/><span class="lab"><font color="Green" title="E93">A56</font></span>: 
<font color="Maroon" title="c20">i</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="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E4:67_3"><span class="lab"><font color="Green" title="E83">A46</font></span></a>, <a class="txt" href="#E5:67_3"><span class="lab"><font color="Green" title="E84">A47</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E15:67_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">R</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:67"><span class="lab"><font color="Green" title="E73">A31</font></span></a>, <a class="txt" href="#E13:67_3"><span class="lab"><font color="Green" title="E92">A55</font></span></a>, <a class="txt" href="#E11:67_3"><span class="lab"><font color="Green" title="E90">A53</font></span></a>, <a class="txt" href="#E6:67_3"><span class="lab"><font color="Green" title="E85">A48</font></span></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="xxreal_2.html#D12" title="XXREAL_2:def.12">XXREAL_2:def 12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">W</font> = <font color="Maroon" title="c12">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c21">R</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></span>;<br/>
<a NAME="E18:67_3"/><span class="lab"><font color="Green" title="E94">A57</font></span>: 
<font color="Maroon" title="c18">Pa</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="jordan5a.html#T1" title="JORDAN5A:th.1">JORDAN5A:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">Pa</font> = <font color="Maroon" title="c18">Pa</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="lab"><font color="Green" title="E95">A58</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c1">P</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">r1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1;<br/>
<a NAME="E21:67_3"/><span class="lab"><font color="Green" title="E96">A59</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">W1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">W1</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">x</font>, <font color="Maroon" title="c26">y</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#K14" title="EUCLID:func.14">Euclid</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="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">W1</font> &amp; <font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">W1</font> implies  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c26">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:67_3_2"/><span class="lab"><font color="Green" title="E97">A60</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">W1</font>
 <span class="kw">and </span><br/><a NAME="E2:67_3_2"/><span class="lab"><font color="Green" title="E98">A61</font></span>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">W1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c26">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c27">pw1</font> = <font color="Maroon" title="c25">x</font>, <font color="Maroon" title="c28">pw2</font> = <font color="Maroon" title="c26">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_3_2"><span class="lab"><font color="Green" title="E97">A60</font></span></a>, <a class="txt" href="#E2:67_3_2"><span class="lab"><font color="Green" title="E98">A61</font></span></a></span>;<br/>
<a NAME="E4:67_3_2"/><span class="lab"><font color="Green" title="E99">A62</font></span>: 
(  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">pw2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">pw2</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>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67_3"><span class="lab"><font color="Green" title="E94">A57</font></span></a>, <a class="txt" href="#E2:67_3_2"><span class="lab"><font color="Green" title="E98">A61</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E5:67_3_2"/>
(  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">pw1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">pw1</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>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67_3"><span class="lab"><font color="Green" title="E94">A57</font></span></a>, <a class="txt" href="#E1:67_3_2"><span class="lab"><font color="Green" title="E97">A60</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E6:67_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A63</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">pw1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">pw2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_3_2"><span class="lab"><font color="Green" title="E99">A62</font></span></a>, <a class="ref" href="jgraph_1.html#T31" target="_self" title="JGRAPH_1:th.31">Th31</a></span>;<br/>
<a NAME="E7:67_3_2"/><span class="lab"><font color="Green" title="E101">A64</font></span>: 
(  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">pw2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c28">pw2</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>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67_3"><span class="lab"><font color="Green" title="E94">A57</font></span></a>, <a class="txt" href="#E2:67_3_2"><span class="lab"><font color="Green" title="E98">A61</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E8:67_3_2"/>
(  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">pw1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">pw1</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>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67_3"><span class="lab"><font color="Green" title="E94">A57</font></span></a>, <a class="txt" href="#E1:67_3_2"><span class="lab"><font color="Green" title="E97">A60</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E9:67_3_2"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">pw1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">pw2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:67_3_2"><span class="lab"><font color="Green" title="E101">A64</font></span></a>, <a class="ref" href="jgraph_1.html#T31" target="_self" title="JGRAPH_1:th.31">Th31</a></span>;<br/>
<a NAME="E10:67_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A65</font></span>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c27">pw1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">pw2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c27">pw1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">pw2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_3_2"><span class="lab"><font color="Green" title="E100">A63</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E11:67_3_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<a NAME="E12:67_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A66</font></span>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c27">pw1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">pw2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c27">pw1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">pw2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67_3_2"><span class="lab"><font color="Green" title="E102">A65</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:67_3_2"/>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">pw1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c28">pw2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">pw1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c28">pw2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c27">pw1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">pw2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c27">pw1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c28">pw2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_1.html#T45" target="_self" title="JGRAPH_1:th.45">Th45</a>, <a class="ref" href="jgraph_1.html#T49" target="_self" title="JGRAPH_1:th.49">Th49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:67_3_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c26">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:67_3_2"><span class="lab"><font color="Green" title="E103">A66</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E22:67_3"/><span class="lab"><font color="Green" title="E97">A67</font></span>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Pa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></span>;<br/>
<a NAME="E23:67_3"/><span class="kw">then </span>
(  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c3">p1</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>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67_3"><span class="lab"><font color="Green" title="E94">A57</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E24:67_3"/><span class="kw">then </span>
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</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="E25:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A68</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E26:67_3"/>
(  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <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>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67_3"><span class="lab"><font color="Green" title="E94">A57</font></span></a>, <a class="txt" href="#E22:67_3"><span class="lab"><font color="Green" title="E97">A67</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E27:67_3"/><span class="kw">then </span>
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</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="E28:67_3"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c23">Pa</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E29:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A69</font></span>: 
<font color="Maroon" title="c19">W1</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:67_3"><span class="lab"><font color="Green" title="E98">A68</font></span></a>, <a class="txt" href="#E21:67_3"><span class="lab"><font color="Green" title="E96">A59</font></span></a>, <a class="ref" href="tbsp_1.html#D9" title="TBSP_1:def.9">TBSP_1:def 9</a></span>;<br/>
<span class="lab"><font color="Green" title="E100">A70</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">f1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 

;<br/>
<a NAME="E31:67_3"/>
<font color="Maroon" title="c20">i</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="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E4:67_3"><span class="lab"><font color="Green" title="E83">A46</font></span></a>, <a class="txt" href="#E5:67_3"><span class="lab"><font color="Green" title="E84">A47</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E32:67_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E33:67_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">h19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E44:67"><span class="lab"><font color="Green" title="E73">A31</font></span></a>, <a class="txt" href="#E8:67_3"><span class="lab"><font color="Green" title="E87">A50</font></span></a>, <a class="txt" href="#E30:67_3"><span class="lab"><font color="Green" title="E100">A70</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E34:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A71</font></span>: 
<font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:67"><span class="lab"><font color="Green" title="E77">A35</font></span></a>, <a class="txt" href="#E7:67_3"><span class="lab"><font color="Green" title="E86">A49</font></span></a>, <a class="txt" href="#E20:67_3"><span class="lab"><font color="Green" title="E95">A58</font></span></a></span>;<br/>
<a NAME="E35:67_3"/><span class="lab"><font color="Green" title="E102">A72</font></span>: 
<font color="Maroon" title="c20">i</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="c16">h19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_3"><span class="lab"><font color="Green" title="E82">A45</font></span></a>, <a class="txt" href="#E10:67_3"><span class="lab"><font color="Green" title="E89">A52</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E36:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A73</font></span>: 
<font color="Maroon" title="c16">h19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:67"><span class="lab"><font color="Green" title="E77">A35</font></span></a></span>;<br/>
<a NAME="E37:67_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">h19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E44:67"><span class="lab"><font color="Green" title="E73">A31</font></span></a>, <a class="txt" href="#E13:67_3"><span class="lab"><font color="Green" title="E92">A55</font></span></a>, <a class="txt" href="#E30:67_3"><span class="lab"><font color="Green" title="E100">A70</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E38:67_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:67"><span class="lab"><font color="Green" title="E77">A35</font></span></a>, <a class="txt" href="#E35:67_3"><span class="lab"><font color="Green" title="E102">A72</font></span></a>, <a class="txt" href="#E20:67_3"><span class="lab"><font color="Green" title="E95">A58</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">w1</font> = <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>, <font color="Maroon" title="c26">w2</font> = <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:67_3"><span class="lab"><font color="Green" title="E101">A71</font></span></a>, <a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></span>;<br/>
<a NAME="E40:67_3"/>
<font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E41:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A74</font></span>: 
<font color="Maroon" title="c15">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:67"><span class="lab"><font color="Green" title="E74">A32</font></span></a>, <a class="txt" href="#E12:67_3"><span class="lab"><font color="Green" title="E91">A54</font></span></a>, <a class="txt" href="#E11:67_3"><span class="lab"><font color="Green" title="E90">A53</font></span></a>, <a class="txt" href="#E14:67_3"><span class="lab"><font color="Green" title="E93">A56</font></span></a>, <a class="txt" href="#E6:67_3"><span class="lab"><font color="Green" title="E85">A48</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E42:67_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:67_3"><span class="lab"><font color="Green" title="E90">A53</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E43:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A75</font></span>: 
<font color="Maroon" title="c25">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c21">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:67_3"><span class="lab"><font color="Green" title="E100">A70</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E44:67_3"/>
<font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_3"><span class="lab"><font color="Green" title="E85">A48</font></span></a>, <a class="txt" href="#E41:67_3"><span class="lab"><font color="Green" title="E104">A74</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E45:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A76</font></span>: 
<font color="Maroon" title="c26">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c21">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:67_3"><span class="lab"><font color="Green" title="E100">A70</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E46:67_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E47:67_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E20:67_3"><span class="lab"><font color="Green" title="E95">A58</font></span></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></span>;<br/>
<a NAME="E48:67_3"/><span class="kw">then </span>
<font color="Maroon" title="c22">W</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:67_3"><span class="lab"><font color="Green" title="E99">A69</font></span></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a>, <a class="ref" href="tbsp_1.html#T21" title="TBSP_1:th.21">TBSP_1:21</a></span>;<br/>
<a NAME="E49:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A77</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c25">w1</font>,<font color="Maroon" title="c26">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c22">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:67_3"><span class="lab"><font color="Green" title="E105">A75</font></span></a>, <a class="txt" href="#E45:67_3"><span class="lab"><font color="Green" title="E106">A76</font></span></a>, <a class="ref" href="tbsp_1.html#D10" title="TBSP_1:def.10">TBSP_1:def 10</a></span>;<br/>
<a NAME="E50:67_3"/>
 <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c22">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:67"><span class="lab"><font color="Green" title="E75">A33</font></span></a>, <a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E3:67_3"><span class="lab"><font color="Green" title="E82">A45</font></span></a>, <a class="txt" href="#E10:67_3"><span class="lab"><font color="Green" title="E89">A52</font></span></a></span>;<br/>
<a NAME="E51:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A78</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c25">w1</font>,<font color="Maroon" title="c26">w2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67_3"><span class="lab"><font color="Green" title="E107">A77</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E52:67_3"/>
<font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_3"><span class="lab"><font color="Green" title="E82">A45</font></span></a>, <a class="txt" href="#E10:67_3"><span class="lab"><font color="Green" title="E89">A52</font></span></a>, <a class="txt" href="#E36:67_3"><span class="lab"><font color="Green" title="E103">A73</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<span class="kw">hence </span><a NAME="E53:67_3"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67_3"><span class="lab"><font color="Green" title="E88">A51</font></span></a>, <a class="txt" href="#E51:67_3"><span class="lab"><font color="Green" title="E108">A78</font></span></a>, <a class="ref" href="jgraph_1.html#T45" target="_self" title="JGRAPH_1:th.45">Th45</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="E57:67"/><span class="lab"><font color="Green" title="E82">A79</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</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="c17">h1</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:67_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">h</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="c16">h19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E54:67"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E2:67_4"/><span class="kw">then </span>
<font color="Maroon" title="c17">h1</font> <a href="funct_1.html#K1" title="FUNCT_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="c17">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">h</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E50:67"><span class="lab"><font color="Green" title="E77">A35</font></span></a></span>;<br/>
<a NAME="E3:67_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A80</font></span>: 
<font color="Maroon" title="c17">h1</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">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">h</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:67"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c18">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">i</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="c17">h1</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:67_4"/><span class="lab"><font color="Green" title="E84">A81</font></span>: 
<font color="Maroon" title="c18">i</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="c17">h1</font>
 ; <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"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/>

<a NAME="E5:67_4"/><span class="kw">then </span>
<font color="Maroon" title="c17">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:67"><span class="lab"><font color="Green" title="E77">A35</font></span></a></span>;<br/>
<a NAME="E6:67_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A82</font></span>: 
<font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_4"><span class="lab"><font color="Green" title="E84">A81</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E7:67_4"/>
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E4:67_4"><span class="lab"><font color="Green" title="E84">A81</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:67_4"/><span class="kw">then </span>
<font color="Maroon" title="c18">i</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="c15">h</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="E9:67_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A83</font></span>: 
<font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">f1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 

;<br/>
<a NAME="E11:67_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A84</font></span>: 
<font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E44:67"><span class="lab"><font color="Green" title="E73">A31</font></span></a>, <a class="txt" href="#E6:67_4"><span class="lab"><font color="Green" title="E85">A82</font></span></a>, <a class="txt" href="#E9:67_4"><span class="lab"><font color="Green" title="E86">A83</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E12:67_4"/>
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="c16">h19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:67"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E13:67_4"/><span class="kw">then </span>
<font color="Maroon" title="c17">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:67"><span class="lab"><font color="Green" title="E77">A35</font></span></a></span>;<br/>
<a NAME="E14:67_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A85</font></span>: 
<font color="Maroon" title="c17">h1</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="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:67"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c1">P</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:67_4"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E19:67"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="txt" href="#E20:67"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E41:67"><span class="lab"><font color="Green" title="E70">A28</font></span></a>, <a class="txt" href="#E42:67"><span class="lab"><font color="Green" title="E71">A29</font></span></a>, <a class="txt" href="#E14:67_4"><span class="lab"><font color="Green" title="E88">A85</font></span></a>, <a class="txt" href="#E3:67_4"><span class="lab"><font color="Green" title="E83">A80</font></span></a>, <a class="txt" href="#E11:67_4"><span class="lab"><font color="Green" title="E87">A84</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="E58:67"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</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="c17">h1</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c17">h1</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="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c17">h1</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="c17">h1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">i</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="c17">h1</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="c17">h1</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="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</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="c17">h1</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="c17">h1</font></span>)</span> ) )</span><br/>

<a NAME="E1:67_5"/><span class="lab"><font color="Green" title="E83">A86</font></span>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c17">h1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c17">h1</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="c17">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:67"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="ref" href="jgraph_1.html#T59" target="_self" title="JGRAPH_1:th.59">Th59</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:67_5"/>
<font color="Maroon" title="c18">i</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="c17">h1</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="c17">h1</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="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</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="c17">h1</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="c17">h1</font></span>)</span> )</span><br/>

<a NAME="E3:67_5"/><span class="kw">then </span>
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c17">h1</font></span>)</span></span>)</span>
 
<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="E4:67_5"/><span class="kw">then </span>
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:67"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="ref" href="jgraph_1.html#T59" target="_self" title="JGRAPH_1:th.59">Th59</a></span>;<br/>
<a NAME="E5:67_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A87</font></span>: 
<font color="Maroon" title="c18">i</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="c17">h1</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="E6:67_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="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_1.html#T61" target="_self" title="JGRAPH_1:th.61">Th61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:67_5"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c17">h1</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="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</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="c17">h1</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="c17">h1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:67"><span class="lab"><font color="Green" title="E82">A79</font></span></a>, <a class="txt" href="#E5:67_5"><span class="lab"><font color="Green" title="E84">A87</font></span></a>, <a class="txt" href="#E1:67_5"><span class="lab"><font color="Green" title="E83">A86</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="E59:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A88</font></span>: 
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c17">h1</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="c17">h1</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="c17">h1</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="c17">h1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard4.html#D2" title="GOBOARD4:def.2">GOBOARD4:def 2</a></span>;<br/>
<a NAME="E60:67"/><span class="lab"><font color="Green" title="E84">A89</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</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="c17">h1</font> holds <br/>( <font color="Maroon" title="c5">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"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <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>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">i</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="c17">h1</font> implies ( <font color="Maroon" title="c5">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"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <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>  ) )</span><br/>

<span class="lab"><font color="Green" title="E85">A90</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c1">P</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>
;<br/>
<span class="kw">assume </span><a NAME="E2:67_6"/><span class="lab"><font color="Green" title="E86">A91</font></span>: 
<font color="Maroon" title="c18">i</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="c17">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">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"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <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>  )</span><br/>

<a NAME="E3:67_6"/><span class="kw">then </span>
<font color="Maroon" title="c17">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:67"><span class="lab"><font color="Green" title="E77">A35</font></span></a></span>;<br/>
<a NAME="E4:67_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A92</font></span>: 
<font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_6"><span class="lab"><font color="Green" title="E86">A91</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E5:67_6"/>
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_6"><span class="lab"><font color="Green" title="E86">A91</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:67_6"/><span class="kw">then </span>
<font color="Maroon" title="c18">i</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="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E7:67_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A93</font></span>: 
<font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">f1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 

;<br/>
<a NAME="E9:67_6"/><span class="kw">then </span>
<font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E44:67"><span class="lab"><font color="Green" title="E73">A31</font></span></a>, <a class="txt" href="#E4:67_6"><span class="lab"><font color="Green" title="E87">A92</font></span></a>, <a class="txt" href="#E7:67_6"><span class="lab"><font color="Green" title="E88">A93</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:67_6"/>
( <font color="Maroon" title="c5">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"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="txt" href="#E1:67_6"><span class="lab"><font color="Green" title="E85">A90</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="E61:67"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</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="c17">h1</font></span>)</span> holds <br/>( <font color="Maroon" title="c5">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="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <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>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">i</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="c17">h1</font></span>)</span> implies ( <font color="Maroon" title="c5">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="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <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>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:67_7"/>
<font color="Maroon" title="c18">i</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="c17">h1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">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="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <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>  )</span><br/>

<a NAME="E2:67_7"/><span class="kw">then </span>
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c17">h1</font></span>)</span></span>)</span>
 
<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="E3:67_7"/><span class="kw">then </span>
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:67"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="ref" href="jgraph_1.html#T60" target="_self" title="JGRAPH_1:th.60">Th60</a></span>;<br/>
<a NAME="E4:67_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A94</font></span>: 
<font color="Maroon" title="c18">i</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="c17">h1</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="E5:67_7"/><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="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_1.html#T61" target="_self" title="JGRAPH_1:th.61">Th61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:67_7"/>
( <font color="Maroon" title="c5">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="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:67"><span class="lab"><font color="Green" title="E84">A89</font></span></a>, <a class="txt" href="#E4:67_7"><span class="lab"><font color="Green" title="E85">A94</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="E62:67"/><span class="kw">then </span>
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c17">h1</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <font color="Maroon" title="c5">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</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="ref" href="goboard4.html#D2" title="GOBOARD4:def.2">GOBOARD4:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">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="E64:67"/><span class="lab"><font color="Green" title="E85">A95</font></span>: 
<font color="Maroon" title="c18">f2</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">and </span><br/><a NAME="E65:67"/><span class="lab"><font color="Green" title="E86">A96</font></span>: 
<font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">and </span><br/><a NAME="E66:67"/><span class="lab"><font color="Green" title="E87">A97</font></span>: 
<font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_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="c18">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">h1</font> <a href="funct_1.html#K1" title="FUNCT_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="c17">h1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E67:67"/><span class="lab"><font color="Green" title="E88">A98</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font>
 <span class="kw">and </span><br/><a NAME="E68:67"/><span class="lab"><font color="Green" title="E89">A99</font></span>: 
(  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c18">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="c17">h1</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="c17">h1</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="c17">h1</font></span>)</span> &amp;  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c18">f2</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <font color="Maroon" title="c5">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> ,<font color="Maroon" title="c6">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">and </span><br/><a NAME="E69:67"/><span class="lab"><font color="Green" title="E90">A100</font></span>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</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="c18">f2</font> holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b2">k</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="c17">h1</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5 )
 <span class="kw">and </span><br/><a NAME="E70:67"/><span class="lab"><font color="Green" title="E91">A101</font></span>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f2</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:67"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="txt" href="#E56:67"><span class="lab"><font color="Green" title="E81">A44</font></span></a>, <a class="txt" href="#E54:67"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="txt" href="#E59:67"><span class="lab"><font color="Green" title="E83">A88</font></span></a>, <a class="ref" href="jgraph_1.html#T56" target="_self" title="JGRAPH_1:th.56">Th56</a></span>;<br/>
<a NAME="E71:67"/><span class="lab"><font color="Green" title="E92">A102</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:67"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="txt" href="#E67:67"><span class="lab"><font color="Green" title="E88">A98</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E72:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A103</font></span>: 
<font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_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="c18">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">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="c18">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1</span>)</span>;<br/>
<a NAME="E73:67"/>
 ex <font color="Olive" title="b1">h29</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h29</font> <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="c14">hb</font> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">i</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="Olive" title="b1">h29</font> holds <br/><font color="Olive" title="b1">h29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">i</font>) ) )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">h29</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E75:67"/><span class="lab"><font color="Green" title="E94">A104</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">h29</font> <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="c14">hb</font>
 <span class="kw">and </span><br/><a NAME="E76:67"/><span class="lab"><font color="Green" title="E95">A105</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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="c19">h29</font> holds <br/><font color="Maroon" title="c19">h29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 ;<br/>
<a NAME="E77:67"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">h29</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">h29</font> or <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:67_8"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">h29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:67_8"/><span class="lab"><font color="Green" title="E96">A106</font></span>: 
<font color="Maroon" title="c21">x</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="c19">h29</font>
 <span class="kw">and </span><br/><a NAME="E4:67_8"/><span class="lab"><font color="Green" title="E97">A107</font></span>: 
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">h29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">i</font> = <font color="Maroon" title="c21">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_8"><span class="lab"><font color="Green" title="E96">A106</font></span></a></span>;<br/>
<a NAME="E6:67_8"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">h29</font> <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> <font color="Maroon" title="c19">h29</font></span>)</span>
 
<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="E7:67_8"/><span class="kw">then </span>
<font color="Maroon" title="c22">i</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="c14">hb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="txt" href="#E3:67_8"><span class="lab"><font color="Green" title="E96">A106</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:67_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A108</font></span>: 
<font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">hb</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="lab"><font color="Green" title="E99">A109</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">g1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 

;<br/>
<span class="lab"><font color="Green" title="E100">A110</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c2">Q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>
;<br/>
<a NAME="E11:67_8"/>
<font color="Maroon" title="c19">h29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:67"><span class="lab"><font color="Green" title="E95">A105</font></span></a>, <a class="txt" href="#E3:67_8"><span class="lab"><font color="Green" title="E96">A106</font></span></a></span>;<br/>
<a NAME="E12:67_8"/><span class="kw">then </span>
<font color="Maroon" title="c19">h29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E37:67"><span class="lab"><font color="Green" title="E67">A25</font></span></a>, <a class="txt" href="#E8:67_8"><span class="lab"><font color="Green" title="E98">A108</font></span></a>, <a class="txt" href="#E9:67_8"><span class="lab"><font color="Green" title="E99">A109</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:67_8"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_8"><span class="lab"><font color="Green" title="E97">A107</font></span></a>, <a class="txt" href="#E10:67_8"><span class="lab"><font color="Green" title="E100">A110</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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">h2</font> = <font color="Maroon" title="c19">h29</font> as   <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">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E96">A111</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c1">P</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>
;<br/>
<a NAME="E80:67"/><span class="lab"><font color="Green" title="E97">A112</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c21">Qa</font> = <font color="Maroon" title="c2">Q</font> as   <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> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">W1</font> = <font color="Maroon" title="c2">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c23">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font> implies <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5 )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:67_10"/><span class="lab"><font color="Green" title="E98">A113</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font>
 <span class="kw">and </span><br/><a NAME="E4:67_10"/><span class="lab"><font color="Green" title="E99">A114</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span><br/>

<a NAME="E5:67_10"/><span class="lab"><font color="Green" title="E100">A115</font></span>: 
<font color="Maroon" title="c21">Qa</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="jordan5a.html#T1" title="JORDAN5A:th.1">JORDAN5A:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">Qa</font> = <font color="Maroon" title="c21">Qa</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="lab"><font color="Green" title="E101">A116</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c2">Q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">r1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1;<br/>
<a NAME="E8:67_10"/><span class="lab"><font color="Green" title="E102">A117</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">W1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">W1</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_10_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">x</font>, <font color="Maroon" title="c27">y</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#K14" title="EUCLID:func.14">Euclid</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="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">W1</font> &amp; <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">W1</font> implies  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c27">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:67_10_2"/><span class="lab"><font color="Green" title="E103">A118</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">W1</font>
 <span class="kw">and </span><br/><a NAME="E2:67_10_2"/><span class="lab"><font color="Green" title="E104">A119</font></span>: 
<font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">W1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c27">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c28">pw1</font> = <font color="Maroon" title="c26">x</font>, <font color="Maroon" title="c29">pw2</font> = <font color="Maroon" title="c27">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_10_2"><span class="lab"><font color="Green" title="E103">A118</font></span></a>, <a class="txt" href="#E2:67_10_2"><span class="lab"><font color="Green" title="E104">A119</font></span></a></span>;<br/>
<a NAME="E4:67_10_2"/><span class="lab"><font color="Green" title="E105">A120</font></span>: 
(  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">pw2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c29">pw2</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>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_10"><span class="lab"><font color="Green" title="E100">A115</font></span></a>, <a class="txt" href="#E2:67_10_2"><span class="lab"><font color="Green" title="E104">A119</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E5:67_10_2"/>
(  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">pw1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">pw1</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>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_10"><span class="lab"><font color="Green" title="E100">A115</font></span></a>, <a class="txt" href="#E1:67_10_2"><span class="lab"><font color="Green" title="E103">A118</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E6:67_10_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A121</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">pw1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">pw2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_10_2"><span class="lab"><font color="Green" title="E105">A120</font></span></a>, <a class="ref" href="jgraph_1.html#T31" target="_self" title="JGRAPH_1:th.31">Th31</a></span>;<br/>
<a NAME="E7:67_10_2"/><span class="lab"><font color="Green" title="E107">A122</font></span>: 
(  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">pw2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c29">pw2</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>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_10"><span class="lab"><font color="Green" title="E100">A115</font></span></a>, <a class="txt" href="#E2:67_10_2"><span class="lab"><font color="Green" title="E104">A119</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E8:67_10_2"/>
(  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">pw1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c28">pw1</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>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_10"><span class="lab"><font color="Green" title="E100">A115</font></span></a>, <a class="txt" href="#E1:67_10_2"><span class="lab"><font color="Green" title="E103">A118</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E9:67_10_2"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">pw1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">pw2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:67_10_2"><span class="lab"><font color="Green" title="E107">A122</font></span></a>, <a class="ref" href="jgraph_1.html#T31" target="_self" title="JGRAPH_1:th.31">Th31</a></span>;<br/>
<a NAME="E10:67_10_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A123</font></span>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">pw1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">pw2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">pw1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">pw2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67_10_2"><span class="lab"><font color="Green" title="E106">A121</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E11:67_10_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<a NAME="E12:67_10_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A124</font></span>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">pw1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">pw2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">pw1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">pw2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67_10_2"><span class="lab"><font color="Green" title="E108">A123</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:67_10_2"/>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c27">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">pw1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c29">pw2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">pw1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c29">pw2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">pw1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">pw2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">pw1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">pw2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_1.html#T45" target="_self" title="JGRAPH_1:th.45">Th45</a>, <a class="ref" href="jgraph_1.html#T49" target="_self" title="JGRAPH_1:th.49">Th49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:67_10_2"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c27">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:67_10_2"><span class="lab"><font color="Green" title="E109">A124</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:67_10"/><span class="lab"><font color="Green" title="E103">A125</font></span>: 
<font color="Maroon" title="c5">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Qa</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></span>;<br/>
<a NAME="E10:67_10"/><span class="kw">then </span>
(  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c5">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>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_10"><span class="lab"><font color="Green" title="E100">A115</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E11:67_10"/><span class="kw">then </span>
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</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="E12:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A126</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">S-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E13:67_10"/>
(  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font> <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; <font color="Maroon" title="c5">q1</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>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_10"><span class="lab"><font color="Green" title="E100">A115</font></span></a>, <a class="txt" href="#E9:67_10"><span class="lab"><font color="Green" title="E103">A125</font></span></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></span>;<br/>
<a NAME="E14:67_10"/><span class="kw">then </span>
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</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="E15:67_10"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">E-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">W-bound</a> <font color="Maroon" title="c24">Qa</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E16:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A127</font></span>: 
<font color="Maroon" title="c22">W1</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:67_10"><span class="lab"><font color="Green" title="E104">A126</font></span></a>, <a class="txt" href="#E8:67_10"><span class="lab"><font color="Green" title="E102">A117</font></span></a>, <a class="ref" href="tbsp_1.html#D9" title="TBSP_1:def.9">TBSP_1:def 9</a></span>;<br/>
<span class="lab"><font color="Green" title="E106">A128</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">g1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 

;<br/>
<a NAME="E18:67_10"/><span class="lab"><font color="Green" title="E107">A129</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_10"><span class="lab"><font color="Green" title="E98">A113</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E19:67_10"/><span class="kw">then </span>
<font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">hb</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="txt" href="#E4:67_10"><span class="lab"><font color="Green" title="E99">A114</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E20:67_10"/><span class="kw">then </span>
<font color="Maroon" title="c23">i</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="c14">hb</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="E21:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A130</font></span>: 
<font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">hb</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E22:67_10"/><span class="lab"><font color="Green" title="E109">A131</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_10"><span class="lab"><font color="Green" title="E99">A114</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E23:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A132</font></span>: 
<font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">hb</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="txt" href="#E3:67_10"><span class="lab"><font color="Green" title="E98">A113</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E24:67_10"/><span class="lab"><font color="Green" title="E111">A133</font></span>: 
<font color="Maroon" title="c23">i</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="c19">h29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_10"><span class="lab"><font color="Green" title="E99">A114</font></span></a>, <a class="txt" href="#E18:67_10"><span class="lab"><font color="Green" title="E107">A129</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E25:67_10"/><span class="kw">then </span>
<font color="Maroon" title="c19">h29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:67"><span class="lab"><font color="Green" title="E95">A105</font></span></a></span>;<br/>
<a NAME="E26:67_10"/><span class="kw">then </span>
<font color="Maroon" title="c19">h29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E37:67"><span class="lab"><font color="Green" title="E67">A25</font></span></a>, <a class="txt" href="#E17:67_10"><span class="lab"><font color="Green" title="E106">A128</font></span></a>, <a class="txt" href="#E21:67_10"><span class="lab"><font color="Green" title="E108">A130</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E27:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A134</font></span>: 
<font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:67"><span class="lab"><font color="Green" title="E95">A105</font></span></a>, <a class="txt" href="#E24:67_10"><span class="lab"><font color="Green" title="E111">A133</font></span></a>, <a class="txt" href="#E7:67_10"><span class="lab"><font color="Green" title="E101">A116</font></span></a></span>;<br/>
<a NAME="E28:67_10"/><span class="lab"><font color="Green" title="E113">A135</font></span>: 
<font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">hb</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="txt" href="#E4:67_10"><span class="lab"><font color="Green" title="E99">A114</font></span></a>, <a class="txt" href="#E18:67_10"><span class="lab"><font color="Green" title="E107">A129</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E29:67_10"/>
<font color="Maroon" title="c23">i</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="c19">h29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_10"><span class="lab"><font color="Green" title="E98">A113</font></span></a>, <a class="txt" href="#E22:67_10"><span class="lab"><font color="Green" title="E109">A131</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E30:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A136</font></span>: 
<font color="Maroon" title="c19">h29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:67"><span class="lab"><font color="Green" title="E95">A105</font></span></a></span>;<br/>
<a NAME="E31:67_10"/>
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">hb</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="txt" href="#E3:67_10"><span class="lab"><font color="Green" title="E98">A113</font></span></a>, <a class="txt" href="#E22:67_10"><span class="lab"><font color="Green" title="E109">A131</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E32:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A137</font></span>: 
<font color="Maroon" title="c23">i</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="c14">hb</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="E33:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A138</font></span>: 
<font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">hb</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E34:67_10"/><span class="kw">then </span>
<font color="Maroon" title="c19">h29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E37:67"><span class="lab"><font color="Green" title="E67">A25</font></span></a>, <a class="txt" href="#E30:67_10"><span class="lab"><font color="Green" title="E114">A136</font></span></a>, <a class="txt" href="#E17:67_10"><span class="lab"><font color="Green" title="E106">A128</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">w1</font> = <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span>, <font color="Maroon" title="c27">w2</font> = <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:67_10"><span class="lab"><font color="Green" title="E114">A136</font></span></a>, <a class="txt" href="#E7:67_10"><span class="lab"><font color="Green" title="E101">A116</font></span></a>, <a class="txt" href="#E27:67_10"><span class="lab"><font color="Green" title="E112">A134</font></span></a>, <a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></span>;<br/>
<a NAME="E36:67_10"/>
<font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">hb</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="txt" href="#E4:67_10"><span class="lab"><font color="Green" title="E99">A114</font></span></a>, <a class="txt" href="#E18:67_10"><span class="lab"><font color="Green" title="E107">A129</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E37:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A139</font></span>: 
<font color="Maroon" title="c23">i</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="c14">hb</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="E38:67_10"/><span class="kw">then </span>
<font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">hb</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">R</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:67"><span class="lab"><font color="Green" title="E67">A25</font></span></a>, <a class="txt" href="#E33:67_10"><span class="lab"><font color="Green" title="E116">A138</font></span></a>, <a class="txt" href="#E23:67_10"><span class="lab"><font color="Green" title="E110">A132</font></span></a>, <a class="txt" href="#E28:67_10"><span class="lab"><font color="Green" title="E113">A135</font></span></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="xxreal_2.html#D12" title="XXREAL_2:def.12">XXREAL_2:def 12</a></span>;<br/>
<a NAME="E40:67_10"/>
<font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E41:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A140</font></span>: 
<font color="Maroon" title="c14">hb</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">hb</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:67"><span class="lab"><font color="Green" title="E68">A26</font></span></a>, <a class="txt" href="#E32:67_10"><span class="lab"><font color="Green" title="E115">A137</font></span></a>, <a class="txt" href="#E23:67_10"><span class="lab"><font color="Green" title="E110">A132</font></span></a>, <a class="txt" href="#E37:67_10"><span class="lab"><font color="Green" title="E117">A139</font></span></a>, <a class="txt" href="#E28:67_10"><span class="lab"><font color="Green" title="E113">A135</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E42:67_10"/><span class="kw">then </span>
<font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:67_10"><span class="lab"><font color="Green" title="E110">A132</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E43:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A141</font></span>: 
<font color="Maroon" title="c26">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c28">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:67_10"><span class="lab"><font color="Green" title="E106">A128</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E44:67_10"/>
<font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:67_10"><span class="lab"><font color="Green" title="E113">A135</font></span></a>, <a class="txt" href="#E41:67_10"><span class="lab"><font color="Green" title="E118">A140</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E45:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A142</font></span>: 
<font color="Maroon" title="c27">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c28">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:67_10"><span class="lab"><font color="Green" title="E106">A128</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c29">W</font> = <font color="Maroon" title="c13">g1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c28">R</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></span>;<br/>
<a NAME="E47:67_10"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E48:67_10"/><span class="kw">then </span>
<font color="Maroon" title="c2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E7:67_10"><span class="lab"><font color="Green" title="E101">A116</font></span></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></span>;<br/>
<a NAME="E49:67_10"/><span class="kw">then </span>
<font color="Maroon" title="c29">W</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:67_10"><span class="lab"><font color="Green" title="E105">A127</font></span></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a>, <a class="ref" href="tbsp_1.html#T21" title="TBSP_1:th.21">TBSP_1:21</a></span>;<br/>
<a NAME="E50:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A143</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c26">w1</font>,<font color="Maroon" title="c27">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c29">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:67_10"><span class="lab"><font color="Green" title="E119">A141</font></span></a>, <a class="txt" href="#E45:67_10"><span class="lab"><font color="Green" title="E120">A142</font></span></a>, <a class="ref" href="tbsp_1.html#D10" title="TBSP_1:def.10">TBSP_1:def 10</a></span>;<br/>
<a NAME="E51:67_10"/>
 <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c29">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:67"><span class="lab"><font color="Green" title="E69">A27</font></span></a>, <a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="txt" href="#E3:67_10"><span class="lab"><font color="Green" title="E98">A113</font></span></a>, <a class="txt" href="#E22:67_10"><span class="lab"><font color="Green" title="E109">A131</font></span></a></span>;<br/>
<a NAME="E52:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A144</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c26">w1</font>,<font color="Maroon" title="c27">w2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:67_10"><span class="lab"><font color="Green" title="E121">A143</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E53:67_10"/>
<font color="Maroon" title="c20">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67_10"><span class="lab"><font color="Green" title="E99">A114</font></span></a>, <a class="txt" href="#E18:67_10"><span class="lab"><font color="Green" title="E107">A129</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E54:67_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A145</font></span>: 
<font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:67"><span class="lab"><font color="Green" title="E95">A105</font></span></a>, <a class="txt" href="#E24:67_10"><span class="lab"><font color="Green" title="E111">A133</font></span></a></span>;<br/>
<a NAME="E55:67_10"/>
<font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_10"><span class="lab"><font color="Green" title="E98">A113</font></span></a>, <a class="txt" href="#E22:67_10"><span class="lab"><font color="Green" title="E109">A131</font></span></a>, <a class="txt" href="#E30:67_10"><span class="lab"><font color="Green" title="E114">A136</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<span class="kw">hence </span><a NAME="E56:67_10"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:67_10"><span class="lab"><font color="Green" title="E123">A145</font></span></a>, <a class="txt" href="#E52:67_10"><span class="lab"><font color="Green" title="E122">A144</font></span></a>, <a class="ref" href="jgraph_1.html#T45" target="_self" title="JGRAPH_1:th.45">Th45</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="E81:67"/><span class="lab"><font color="Green" title="E98">A146</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="c14">hb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:67"><span class="lab"><font color="Green" title="E66">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E82:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A147</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">hb</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="c14">hb</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="E83:67"/><span class="lab"><font color="Green" title="E100">A148</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="c14">hb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:67"><span class="lab"><font color="Green" title="E66">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E84:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A149</font></span>: 
<font color="Maroon" title="c20">h2</font> <a href="funct_1.html#K1" title="FUNCT_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="c20">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h2</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="c20">h2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E85:67"/><span class="lab"><font color="Green" title="E102">A150</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">hb</font> <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> <font color="Maroon" title="c14">hb</font></span>)</span>
 
<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="E86:67"/><span class="lab"><font color="Green" title="E103">A151</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</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="c14">hb</font> holds <br/><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">i</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="c14">hb</font> implies <font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:67_11"/><span class="lab"><font color="Green" title="E104">A152</font></span>: 
<font color="Maroon" title="c21">i</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="c14">hb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span></span><br/>

<a NAME="E2:67_11"/><span class="kw">then </span>
<font color="Maroon" title="c21">i</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="c19">h29</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<a NAME="E3:67_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A153</font></span>: 
<font color="Maroon" title="c20">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:67"><span class="lab"><font color="Green" title="E95">A105</font></span></a></span>;<br/>
<a NAME="E4:67_11"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c21">i</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="c14">hb</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:67"><span class="lab"><font color="Green" title="E102">A150</font></span></a>, <a class="txt" href="#E1:67_11"><span class="lab"><font color="Green" title="E104">A152</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:67_11"/>
<font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="txt" href="#E3:67_11"><span class="lab"><font color="Green" title="E105">A153</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</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="E87:67"/><span class="lab"><font color="Green" title="E104">A154</font></span>: 
<font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:67"><span class="lab"><font color="Green" title="E92">A102</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E88:67"/><span class="lab"><font color="Green" title="E105">A155</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="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:67"><span class="lab"><font color="Green" title="E72">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E89:67"/><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="c16">h19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E90:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A156</font></span>: 
<font color="Maroon" title="c17">h1</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="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:67"><span class="lab"><font color="Green" title="E77">A35</font></span></a>, <a class="txt" href="#E55:67"><span class="lab"><font color="Green" title="E80">A43</font></span></a></span>;<br/>
<a NAME="E91:67"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">h</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="c16">h19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E88:67"><span class="lab"><font color="Green" title="E105">A155</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E92:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A157</font></span>: 
<font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">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="c18">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E19:67"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="txt" href="#E20:67"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E41:67"><span class="lab"><font color="Green" title="E70">A28</font></span></a>, <a class="txt" href="#E42:67"><span class="lab"><font color="Green" title="E71">A29</font></span></a>, <a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E50:67"><span class="lab"><font color="Green" title="E77">A35</font></span></a>, <a class="txt" href="#E65:67"><span class="lab"><font color="Green" title="E86">A96</font></span></a>, <a class="txt" href="#E66:67"><span class="lab"><font color="Green" title="E87">A97</font></span></a>, <a class="txt" href="#E55:67"><span class="lab"><font color="Green" title="E80">A43</font></span></a>, <a class="txt" href="#E87:67"><span class="lab"><font color="Green" title="E104">A154</font></span></a>, <a class="txt" href="#E72:67"><span class="lab"><font color="Green" title="E93">A103</font></span></a>, <a class="txt" href="#E90:67"><span class="lab"><font color="Green" title="E106">A156</font></span></a>, <a class="ref" href="jordan6.html#T49" title="JORDAN6:th.49">JORDAN6:49</a></span>;<br/>
<a NAME="E93:67"/>
5 <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">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:67"><span class="lab"><font color="Green" title="E72">A30</font></span></a>, <a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E67:67"><span class="lab"><font color="Green" title="E88">A98</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E94:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A158</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">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="E95:67"/><span class="lab"><font color="Green" title="E109">A159</font></span>: 
<font color="Maroon" title="c17">h1</font> <a href="funct_1.html#K1" title="FUNCT_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="c17">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">h1</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">h1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:67"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E110">A160</font></span>: <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c18">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="c18">f2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:67"><span class="lab"><font color="Green" title="E87">A97</font></span></a>, <a class="txt" href="#E71:67"><span class="lab"><font color="Green" title="E92">A102</font></span></a>, <a class="txt" href="#E72:67"><span class="lab"><font color="Green" title="E93">A103</font></span></a>, <a class="ref" href="jgraph_1.html#T59" target="_self" title="JGRAPH_1:th.59">Th59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c17">h1</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="c17">h1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:67"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="ref" href="jgraph_1.html#T59" target="_self" title="JGRAPH_1:th.59">Th59</a></span>
;<br/>
<a NAME="E97:67"/><span class="lab"><font color="Green" title="E111">A161</font></span>: 
<font color="Maroon" title="c20">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:67"><span class="lab"><font color="Green" title="E100">A148</font></span></a>, <a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E98:67"/><span class="lab"><font color="Green" title="E112">A162</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:67"><span class="lab"><font color="Green" title="E66">A24</font></span></a>, <a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E99:67"/><span class="lab"><font color="Green" title="E113">A163</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</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="c20">h2</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <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"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <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"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">i</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="c20">h2</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <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"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <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"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:67_13"/>
<font color="Maroon" title="c21">i</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="c20">h2</font>
 ; <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"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <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"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <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"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<a NAME="E2:67_13"/><span class="kw">then </span>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font></span>)</span>
 
<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="E3:67_13"/><span class="kw">then </span>
<font color="Maroon" title="c21">i</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="c14">hb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:67_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A164</font></span>: 
( <font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> &amp; <font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">hb</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E86:67"><span class="lab"><font color="Green" title="E103">A151</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">g1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 

;<br/>
<a NAME="E6:67_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A165</font></span>: 
<font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E37:67"><span class="lab"><font color="Green" title="E67">A25</font></span></a>, <a class="txt" href="#E4:67_13"><span class="lab"><font color="Green" title="E114">A164</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E7:67_13"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">hb</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="txt" href="#E98:67"><span class="lab"><font color="Green" title="E112">A162</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E8:67_13"/><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="c14">hb</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="E9:67_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A166</font></span>: 
<font color="Maroon" title="c20">h2</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="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E86:67"><span class="lab"><font color="Green" title="E103">A151</font></span></a></span>;<br/>
<a NAME="E10:67_13"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">hb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">hb</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="txt" href="#E98:67"><span class="lab"><font color="Green" title="E112">A162</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E11:67_13"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">hb</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="c14">hb</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="E12:67_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A167</font></span>: 
<font color="Maroon" title="c20">h2</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="c20">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">hb</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="txt" href="#E86:67"><span class="lab"><font color="Green" title="E103">A151</font></span></a></span>;<br/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c2">Q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E14:67_13"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <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"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <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"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E9:67"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E34:67"><span class="lab"><font color="Green" title="E64">A22</font></span></a>, <a class="txt" href="#E35:67"><span class="lab"><font color="Green" title="E65">A23</font></span></a>, <a class="txt" href="#E9:67_13"><span class="lab"><font color="Green" title="E116">A166</font></span></a>, <a class="txt" href="#E12:67_13"><span class="lab"><font color="Green" title="E117">A167</font></span></a>, <a class="txt" href="#E6:67_13"><span class="lab"><font color="Green" title="E115">A165</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="E100:67"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</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="c20">h2</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c20">h2</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="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</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="c20">h2</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="c20">h2</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">i</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="c20">h2</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="c20">h2</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="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</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="c20">h2</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="c20">h2</font></span>)</span> ) )</span><br/>

<a NAME="E1:67_14"/><span class="lab"><font color="Green" title="E114">A168</font></span>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c20">h2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c20">h2</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="c20">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font></span>)</span></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="#E98:67"><span class="lab"><font color="Green" title="E112">A162</font></span></a>, <a class="ref" href="jgraph_1.html#T60" target="_self" title="JGRAPH_1:th.60">Th60</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:67_14"/>
<font color="Maroon" title="c21">i</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="c20">h2</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="c20">h2</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="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</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="c20">h2</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="c20">h2</font></span>)</span> )</span><br/>

<a NAME="E3:67_14"/><span class="kw">then </span>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c20">h2</font></span>)</span></span>)</span>
 
<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="E4:67_14"/><span class="kw">then </span>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E98:67"><span class="lab"><font color="Green" title="E112">A162</font></span></a>, <a class="ref" href="jgraph_1.html#T60" target="_self" title="JGRAPH_1:th.60">Th60</a></span>;<br/>
<a NAME="E5:67_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A169</font></span>: 
<font color="Maroon" title="c21">i</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="c20">h2</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="E6:67_14"/><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="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_1.html#T61" target="_self" title="JGRAPH_1:th.61">Th61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:67_14"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c20">h2</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="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</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="c20">h2</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="c20">h2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E99:67"><span class="lab"><font color="Green" title="E113">A163</font></span></a>, <a class="txt" href="#E5:67_14"><span class="lab"><font color="Green" title="E115">A169</font></span></a>, <a class="txt" href="#E1:67_14"><span class="lab"><font color="Green" title="E114">A168</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="E101:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A170</font></span>: 
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c20">h2</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="c20">h2</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="c20">h2</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="c20">h2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard4.html#D2" title="GOBOARD4:def.2">GOBOARD4:def 2</a></span>;<br/>
<a NAME="E102:67"/><span class="lab"><font color="Green" title="E115">A171</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</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="c20">h2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">i</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="c20">h2</font> implies ( <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> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) )</span><br/>

<span class="lab"><font color="Green" title="E116">A172</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c2">Q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>
;<br/>
<span class="kw">assume </span><a NAME="E2:67_15"/>
<font color="Maroon" title="c21">i</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="c20">h2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/>

<a NAME="E3:67_15"/><span class="kw">then </span>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font></span>)</span>
 
<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="E4:67_15"/><span class="kw">then </span>
<font color="Maroon" title="c21">i</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="c14">hb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:67_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A173</font></span>: 
( <font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> &amp; <font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">hb</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E86:67"><span class="lab"><font color="Green" title="E103">A151</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">g1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 

;<br/>
<a NAME="E7:67_15"/><span class="kw">then </span>
<font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E37:67"><span class="lab"><font color="Green" title="E67">A25</font></span></a>, <a class="txt" href="#E5:67_15"><span class="lab"><font color="Green" title="E117">A173</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:67_15"/>
( <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> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">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="#E4:67"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E1:67_15"><span class="lab"><font color="Green" title="E116">A172</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="E103:67"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</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="c20">h2</font></span>)</span> 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> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_16"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">i</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="c20">h2</font></span>)</span> implies ( <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> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:67_16"/>
<font color="Maroon" title="c21">i</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="c20">h2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/>

<a NAME="E2:67_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c20">h2</font></span>)</span></span>)</span>
 
<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="E3:67_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E98:67"><span class="lab"><font color="Green" title="E112">A162</font></span></a>, <a class="ref" href="jgraph_1.html#T59" target="_self" title="JGRAPH_1:th.59">Th59</a></span>;<br/>
<a NAME="E4:67_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A174</font></span>: 
<font color="Maroon" title="c21">i</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="c20">h2</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="E5:67_16"/><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="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_1.html#T61" target="_self" title="JGRAPH_1:th.61">Th61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:67_16"/>
( <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> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">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="#E102:67"><span class="lab"><font color="Green" title="E115">A171</font></span></a>, <a class="txt" href="#E4:67_16"><span class="lab"><font color="Green" title="E116">A174</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="E104:67"/><span class="kw">then </span>
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c20">h2</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> ,<font color="Maroon" title="c4">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard4.html#D2" title="GOBOARD4:def.2">GOBOARD4:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">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="E106:67"/><span class="lab"><font color="Green" title="E116">A175</font></span>: 
<font color="Maroon" title="c21">g2</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">and </span><br/><a NAME="E107:67"/><span class="lab"><font color="Green" title="E117">A176</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">and </span><br/><a NAME="E108:67"/><span class="lab"><font color="Green" title="E118">A177</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="funct_1.html#K1" title="FUNCT_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="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h2</font> <a href="funct_1.html#K1" title="FUNCT_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="c20">h2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E109:67"/><span class="lab"><font color="Green" title="E119">A178</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font>
 <span class="kw">and </span><br/><a NAME="E110:67"/><span class="lab"><font color="Green" title="E120">A179</font></span>: 
(  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c21">g2</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="c20">h2</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="c20">h2</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="c20">h2</font></span>)</span> &amp;  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c21">g2</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> ,<font color="Maroon" title="c4">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E111:67"/><span class="lab"><font color="Green" title="E121">A180</font></span>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</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="c21">g2</font> holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b2">k</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="c20">h2</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5 )
 <span class="kw">and </span><br/><a NAME="E112:67"/><span class="lab"><font color="Green" title="E122">A181</font></span>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:67"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="txt" href="#E98:67"><span class="lab"><font color="Green" title="E112">A162</font></span></a>, <a class="txt" href="#E101:67"><span class="lab"><font color="Green" title="E114">A170</font></span></a>, <a class="txt" href="#E80:67"><span class="lab"><font color="Green" title="E97">A112</font></span></a>, <a class="ref" href="jgraph_1.html#T57" target="_self" title="JGRAPH_1:th.57">Th57</a></span>;<br/>
<a NAME="E113:67"/>
5 <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="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:67"><span class="lab"><font color="Green" title="E66">A24</font></span></a>, <a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="txt" href="#E109:67"><span class="lab"><font color="Green" title="E119">A178</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E114:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A182</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="c21">g2</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="E115:67"/><span class="lab"><font color="Green" title="E124">A183</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E98:67"><span class="lab"><font color="Green" title="E112">A162</font></span></a>, <a class="txt" href="#E109:67"><span class="lab"><font color="Green" title="E119">A178</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E116:67"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</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#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E125">A184</font></span>: <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</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="#E107:67"><span class="lab"><font color="Green" title="E117">A176</font></span></a>, <a class="txt" href="#E115:67"><span class="lab"><font color="Green" title="E124">A183</font></span></a>, <a class="txt" href="#E97:67"><span class="lab"><font color="Green" title="E111">A161</font></span></a>, <a class="ref" href="jgraph_1.html#T60" target="_self" title="JGRAPH_1:th.60">Th60</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c20">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E98:67"><span class="lab"><font color="Green" title="E112">A162</font></span></a>, <a class="ref" href="jgraph_1.html#T60" target="_self" title="JGRAPH_1:th.60">Th60</a></span>
;<br/>
<a NAME="E118:67"/>
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="c14">hb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:67"><span class="lab"><font color="Green" title="E98">A146</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E119:67"/><span class="kw">then </span>
<font color="Maroon" title="c20">h2</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="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E86:67"><span class="lab"><font color="Green" title="E103">A151</font></span></a></span>;<br/>
<a NAME="E120:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A185</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">g2</font> <a href="funct_1.html#K1" title="FUNCT_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="c21">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E9:67"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E34:67"><span class="lab"><font color="Green" title="E64">A22</font></span></a>, <a class="txt" href="#E35:67"><span class="lab"><font color="Green" title="E65">A23</font></span></a>, <a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="txt" href="#E86:67"><span class="lab"><font color="Green" title="E103">A151</font></span></a>, <a class="txt" href="#E107:67"><span class="lab"><font color="Green" title="E117">A176</font></span></a>, <a class="txt" href="#E108:67"><span class="lab"><font color="Green" title="E118">A177</font></span></a>, <a class="txt" href="#E97:67"><span class="lab"><font color="Green" title="E111">A161</font></span></a>, <a class="txt" href="#E84:67"><span class="lab"><font color="Green" title="E101">A149</font></span></a>, <a class="txt" href="#E82:67"><span class="lab"><font color="Green" title="E99">A147</font></span></a>, <a class="ref" href="jordan6.html#T49" title="JORDAN6:th.49">JORDAN6:49</a></span>;<br/>
<a NAME="E121:67"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">hb</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="c14">hb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:67"><span class="lab"><font color="Green" title="E100">A148</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E122:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A186</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="funct_1.html#K1" title="FUNCT_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="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E35:67"><span class="lab"><font color="Green" title="E65">A23</font></span></a>, <a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="txt" href="#E86:67"><span class="lab"><font color="Green" title="E103">A151</font></span></a>, <a class="txt" href="#E108:67"><span class="lab"><font color="Green" title="E118">A177</font></span></a>, <a class="txt" href="#E84:67"><span class="lab"><font color="Green" title="E101">A149</font></span></a></span>;<br/>
<a NAME="E123:67"/>
<font color="Maroon" title="c21">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="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="funct_1.html#K1" title="FUNCT_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="c21">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:67"><span class="lab"><font color="Green" title="E124">A183</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E124:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A187</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c21">g2</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="c21">g2</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="#E115:67"><span class="lab"><font color="Green" title="E124">A183</font></span></a>, <a class="txt" href="#E122:67"><span class="lab"><font color="Green" title="E127">A186</font></span></a>, <a class="ref" href="jgraph_1.html#T60" target="_self" title="JGRAPH_1:th.60">Th60</a></span>;<br/>
<a NAME="E125:67"/>
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="c14">hb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:67"><span class="lab"><font color="Green" title="E100">A148</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E126:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A188</font></span>: 
<font color="Maroon" title="c20">h2</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="c5">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:67"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E34:67"><span class="lab"><font color="Green" title="E64">A22</font></span></a>, <a class="txt" href="#E86:67"><span class="lab"><font color="Green" title="E103">A151</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E130">A189</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">g</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c2">Q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>
;<br/>
<a NAME="E128:67"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">h</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="c16">h19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E54:67"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E129:67"/><span class="kw">then </span>
<font color="Maroon" title="c17">h1</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">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:67"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E42:67"><span class="lab"><font color="Green" title="E71">A29</font></span></a>, <a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E50:67"><span class="lab"><font color="Green" title="E77">A35</font></span></a>, <a class="txt" href="#E95:67"><span class="lab"><font color="Green" title="E109">A159</font></span></a></span>;<br/>
<a NAME="E130:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A190</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c18">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="c18">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="#E66:67"><span class="lab"><font color="Green" title="E87">A97</font></span></a>, <a class="txt" href="#E71:67"><span class="lab"><font color="Green" title="E92">A102</font></span></a>, <a class="txt" href="#E95:67"><span class="lab"><font color="Green" title="E109">A159</font></span></a>, <a class="txt" href="#E72:67"><span class="lab"><font color="Green" title="E93">A103</font></span></a>, <a class="ref" href="jgraph_1.html#T59" target="_self" title="JGRAPH_1:th.59">Th59</a></span>;<br/>
<a NAME="E131:67"/>
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="c16">h19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:67"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E132:67"/><span class="kw">then </span>
<font color="Maroon" title="c17">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:67"><span class="lab"><font color="Green" title="E77">A35</font></span></a></span>;<br/>
<a NAME="E133:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A191</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c18">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="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="#E19:67"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E41:67"><span class="lab"><font color="Green" title="E70">A28</font></span></a>, <a class="txt" href="#E65:67"><span class="lab"><font color="Green" title="E86">A96</font></span></a>, <a class="txt" href="#E71:67"><span class="lab"><font color="Green" title="E92">A102</font></span></a>, <a class="txt" href="#E87:67"><span class="lab"><font color="Green" title="E104">A154</font></span></a>, <a class="ref" href="jgraph_1.html#T59" target="_self" title="JGRAPH_1:th.59">Th59</a></span>;<br/>
<a NAME="E134:67"/>
<font color="Maroon" title="c21">g2</font> <a href="funct_1.html#K1" title="FUNCT_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="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">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="c21">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:67"><span class="lab"><font color="Green" title="E124">A183</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E133">A192</font></span>: <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c21">g2</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="c21">g2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font></span>)</span></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="#E108:67"><span class="lab"><font color="Green" title="E118">A177</font></span></a>, <a class="txt" href="#E115:67"><span class="lab"><font color="Green" title="E124">A183</font></span></a>, <a class="txt" href="#E84:67"><span class="lab"><font color="Green" title="E101">A149</font></span></a>, <a class="ref" href="jgraph_1.html#T60" target="_self" title="JGRAPH_1:th.60">Th60</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c20">h2</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="c20">h2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E98:67"><span class="lab"><font color="Green" title="E112">A162</font></span></a>, <a class="ref" href="jgraph_1.html#T60" target="_self" title="JGRAPH_1:th.60">Th60</a></span>
;<br/>
<a NAME="E136:67"/>
<font color="Maroon" title="c21">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="c21">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E115:67"><span class="lab"><font color="Green" title="E124">A183</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E137:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A193</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c21">g2</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="c5">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="#E107:67"><span class="lab"><font color="Green" title="E117">A176</font></span></a>, <a class="txt" href="#E115:67"><span class="lab"><font color="Green" title="E124">A183</font></span></a>, <a class="txt" href="#E126:67"><span class="lab"><font color="Green" title="E129">A188</font></span></a>, <a class="txt" href="#E97:67"><span class="lab"><font color="Green" title="E111">A161</font></span></a>, <a class="ref" href="jgraph_1.html#T60" target="_self" title="JGRAPH_1:th.60">Th60</a></span>;<br/>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c18">f2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:67"><span class="lab"><font color="Green" title="E86">A96</font></span></a>, <a class="txt" href="#E71:67"><span class="lab"><font color="Green" title="E92">A102</font></span></a>, <a class="txt" href="#E55:67"><span class="lab"><font color="Green" title="E80">A43</font></span></a>, <a class="txt" href="#E87:67"><span class="lab"><font color="Green" title="E104">A154</font></span></a>, <a class="ref" href="jgraph_1.html#T59" target="_self" title="JGRAPH_1:th.59">Th59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c17">h1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:67"><span class="lab"><font color="Green" title="E79">A42</font></span></a>, <a class="ref" href="jgraph_1.html#T59" target="_self" title="JGRAPH_1:th.59">Th59</a></span>
;<br/>
<a NAME="E139:67"/><span class="kw">then </span>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c18">f2</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="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:67"><span class="lab"><font color="Green" title="E85">A95</font></span></a>, <a class="txt" href="#E68:67"><span class="lab"><font color="Green" title="E89">A99</font></span></a>, <a class="txt" href="#E106:67"><span class="lab"><font color="Green" title="E116">A175</font></span></a>, <a class="txt" href="#E110:67"><span class="lab"><font color="Green" title="E120">A179</font></span></a>, <a class="txt" href="#E87:67"><span class="lab"><font color="Green" title="E104">A154</font></span></a>, <a class="txt" href="#E72:67"><span class="lab"><font color="Green" title="E93">A103</font></span></a>, <a class="txt" href="#E133:67"><span class="lab"><font color="Green" title="E132">A191</font></span></a>, <a class="txt" href="#E130:67"><span class="lab"><font color="Green" title="E131">A190</font></span></a>, <a class="txt" href="#E137:67"><span class="lab"><font color="Green" title="E134">A193</font></span></a>, <a class="txt" href="#E124:67"><span class="lab"><font color="Green" title="E128">A187</font></span></a>, <a class="txt" href="#E96:67"><span class="lab"><font color="Green" title="E110">A160</font></span></a>, <a class="txt" href="#E117:67"><span class="lab"><font color="Green" title="E125">A184</font></span></a>, <a class="txt" href="#E135:67"><span class="lab"><font color="Green" title="E133">A192</font></span></a>, <a class="txt" href="#E94:67"><span class="lab"><font color="Green" title="E108">A158</font></span></a>, <a class="txt" href="#E114:67"><span class="lab"><font color="Green" title="E123">A182</font></span></a>, <a class="txt" href="#E92:67"><span class="lab"><font color="Green" title="E107">A157</font></span></a>, <a class="txt" href="#E120:67"><span class="lab"><font color="Green" title="E126">A185</font></span></a>, <a class="ref" href="jgraph_1.html#T30" target="_self" title="JGRAPH_1:th.30">Th30</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E141:67"/><span class="lab"><font color="Green" title="E135">A194</font></span>: 
<font color="Maroon" title="c22">s</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="c18">f2</font>
 <span class="kw">and </span><br/><a NAME="E142:67"/><span class="lab"><font color="Green" title="E136">A195</font></span>: 
<font color="Maroon" title="c22">s</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="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">ps</font> = <font color="Maroon" title="c22">s</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E141:67"><span class="lab"><font color="Green" title="E135">A194</font></span></a></span>;<br/>
<a NAME="E144:67"/>
<font color="Maroon" title="c23">ps</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E142:67"><span class="lab"><font color="Green" title="E136">A195</font></span></a>, <a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E146:67"/><span class="lab"><font color="Green" title="E137">A196</font></span>: 
( <font color="Maroon" title="c23">ps</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">y</font> &amp; <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> ) </span>}</span>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E147:67"/>
<font color="Maroon" title="c23">ps</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c18">f2</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f2</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E141:67"><span class="lab"><font color="Green" title="E135">A194</font></span></a>, <a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E149:67"/><span class="lab"><font color="Green" title="E138">A197</font></span>: 
( <font color="Maroon" title="c23">ps</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c18">f2</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f2</font> ) </span>}</span>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E151:67"/><span class="lab"><font color="Green" title="E139">A198</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c18">f2</font>,<font color="Maroon" title="c26">i</font>
 <span class="kw">and </span><br/><a NAME="E152:67"/><span class="lab"><font color="Green" title="E140">A199</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">i</font>
 <span class="kw">and </span><br/><a NAME="E153:67"/><span class="lab"><font color="Green" title="E141">A200</font></span>: 
<font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E149:67"><span class="lab"><font color="Green" title="E138">A197</font></span></a></span>;<br/>
<a NAME="E154:67"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c18">f2</font>,<font color="Maroon" title="c26">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E152:67"><span class="lab"><font color="Green" title="E140">A199</font></span></a>, <a class="txt" href="#E153:67"><span class="lab"><font color="Green" title="E141">A200</font></span></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></span>;<br/>
<a NAME="E155:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A201</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">ps</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">i</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E149:67"><span class="lab"><font color="Green" title="E138">A197</font></span></a>, <a class="txt" href="#E151:67"><span class="lab"><font color="Green" title="E139">A198</font></span></a>, <a class="ref" href="jgraph_1.html#T53" target="_self" title="JGRAPH_1:th.53">Th53</a></span>;<br/>
<a NAME="E156:67"/>
<font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E153:67"><span class="lab"><font color="Green" title="E141">A200</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E157:67"/><span class="kw">then </span>
<font color="Maroon" title="c26">i</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="c18">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E152:67"><span class="lab"><font color="Green" title="E140">A199</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E159:67"/><span class="lab"><font color="Green" title="E143">A202</font></span>: 
<font color="Maroon" title="c27">k</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="c17">h1</font>
 <span class="kw">and </span><br/><a NAME="E160:67"/><span class="lab"><font color="Green" title="E144">A203</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">k</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:67"><span class="lab"><font color="Green" title="E90">A100</font></span></a></span>;<br/>
<a NAME="E161:67"/>
<font color="Maroon" title="c27">k</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="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:67"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="txt" href="#E159:67"><span class="lab"><font color="Green" title="E143">A202</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<a NAME="E162:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E145">A204</font></span>: 
<font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c28">p11</font> = <font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">k</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="E164:67"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:67"><span class="lab"><font color="Green" title="E91">A101</font></span></a>, <a class="txt" href="#E152:67"><span class="lab"><font color="Green" title="E140">A199</font></span></a>, <a class="txt" href="#E153:67"><span class="lab"><font color="Green" title="E141">A200</font></span></a></span>;<br/>
<a NAME="E165:67"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">ps</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">i</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E155:67"><span class="lab"><font color="Green" title="E142">A201</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E166:67"/><span class="kw">then </span>
( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">ps</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">k</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">ps</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">i</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">k</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">ps</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">i</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">i</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">k</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E160:67"><span class="lab"><font color="Green" title="E144">A203</font></span></a>, <a class="ref" href="toprns_1.html#T35" title="TOPRNS_1:th.35">TOPRNS_1:35</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E167:67"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">ps</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">k</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span>
 
<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="E168:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E146">A205</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">p11</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c23">ps</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprns_1.html#T28" title="TOPRNS_1:th.28">TOPRNS_1:28</a></span>;<br/>
<a NAME="E169:67"/>
<font color="Maroon" title="c27">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E159:67"><span class="lab"><font color="Green" title="E143">A202</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E170:67"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">k</font> &amp; <font color="Maroon" title="c27">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">h1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E171:67"/><span class="kw">then </span>
<font color="Maroon" title="c17">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E172:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E147">A206</font></span>: 
<font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c27">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:67"><span class="lab"><font color="Green" title="E77">A35</font></span></a>, <a class="txt" href="#E159:67"><span class="lab"><font color="Green" title="E143">A202</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c29">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E174:67"/><span class="lab"><font color="Green" title="E148">A207</font></span>: 
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c29">j</font>
 <span class="kw">and </span><br/><a NAME="E175:67"/><span class="lab"><font color="Green" title="E149">A208</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">j</font>
 <span class="kw">and </span><br/><a NAME="E176:67"/><span class="lab"><font color="Green" title="E150">A209</font></span>: 
<font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E146:67"><span class="lab"><font color="Green" title="E137">A196</font></span></a></span>;<br/>
<a NAME="E177:67"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> <font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c29">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E175:67"><span class="lab"><font color="Green" title="E149">A208</font></span></a>, <a class="txt" href="#E176:67"><span class="lab"><font color="Green" title="E150">A209</font></span></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></span>;<br/>
<a NAME="E178:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E151">A210</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">ps</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">j</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">j</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E146:67"><span class="lab"><font color="Green" title="E137">A196</font></span></a>, <a class="txt" href="#E174:67"><span class="lab"><font color="Green" title="E148">A207</font></span></a>, <a class="ref" href="jgraph_1.html#T53" target="_self" title="JGRAPH_1:th.53">Th53</a></span>;<br/>
<a NAME="E179:67"/>
<font color="Maroon" title="c29">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E176:67"><span class="lab"><font color="Green" title="E150">A209</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E180:67"/><span class="kw">then </span>
<font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E175:67"><span class="lab"><font color="Green" title="E149">A208</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E181:67"/><span class="kw">then </span>
<font color="Maroon" title="c29">j</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="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c30">k9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E183:67"/><span class="lab"><font color="Green" title="E152">A211</font></span>: 
<font color="Maroon" title="c30">k9</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="c20">h2</font>
 <span class="kw">and </span><br/><a NAME="E184:67"/><span class="lab"><font color="Green" title="E153">A212</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">j</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c30">k9</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E111:67"><span class="lab"><font color="Green" title="E121">A180</font></span></a></span>;<br/>
<a NAME="E185:67"/>
<font color="Maroon" title="c30">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">h2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E183:67"><span class="lab"><font color="Green" title="E152">A211</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E186:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E154">A213</font></span>: 
<font color="Maroon" title="c30">k9</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="c14">hb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:67"><span class="lab"><font color="Green" title="E94">A104</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E187:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E155">A214</font></span>: 
<font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c30">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">hb</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c31">q11</font> = <font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c30">k9</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="E189:67"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">j</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E112:67"><span class="lab"><font color="Green" title="E122">A181</font></span></a>, <a class="txt" href="#E175:67"><span class="lab"><font color="Green" title="E149">A208</font></span></a>, <a class="txt" href="#E176:67"><span class="lab"><font color="Green" title="E150">A209</font></span></a></span>;<br/>
<a NAME="E190:67"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">ps</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">j</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E178:67"><span class="lab"><font color="Green" title="E151">A210</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E191:67"/><span class="kw">then </span>
( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">ps</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c30">k9</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">ps</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">j</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">j</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c30">k9</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">ps</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">j</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">j</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c30">k9</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E184:67"><span class="lab"><font color="Green" title="E153">A212</font></span></a>, <a class="ref" href="toprns_1.html#T35" title="TOPRNS_1:th.35">TOPRNS_1:35</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E192:67"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">ps</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c30">k9</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span>
 
<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="E193:67"/><span class="kw">then </span>
( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">p11</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c31">q11</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">p11</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c23">ps</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">ps</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c31">q11</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">p11</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c23">ps</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">ps</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c31">q11</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E168:67"><span class="lab"><font color="Green" title="E146">A205</font></span></a>, <a class="ref" href="toprns_1.html#T35" title="TOPRNS_1:th.35">TOPRNS_1:35</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E194:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E156">A215</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">p11</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c31">q11</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span>
 
<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="E195:67"/>
<font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c30">k9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">hb</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c30">k9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E86:67"><span class="lab"><font color="Green" title="E103">A151</font></span></a>, <a class="txt" href="#E186:67"><span class="lab"><font color="Green" title="E154">A213</font></span></a></span>;<br/>
<a NAME="E196:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A216</font></span>: 
<font color="Maroon" title="c20">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c30">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:67"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E37:67"><span class="lab"><font color="Green" title="E67">A25</font></span></a>, <a class="txt" href="#E187:67"><span class="lab"><font color="Green" title="E155">A214</font></span></a>, <a class="txt" href="#E51:67"><span class="lab"><font color="Green" title="E78">A36</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c32">x1</font> = <font color="Maroon" title="c28">p11</font>, <font color="Maroon" title="c33">x2</font> = <font color="Maroon" title="c31">q11</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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">f1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:67"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 

;<br/>
<a NAME="E199:67"/><span class="kw">then </span>
<font color="Maroon" title="c17">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E44:67"><span class="lab"><font color="Green" title="E73">A31</font></span></a>, <a class="txt" href="#E172:67"><span class="lab"><font color="Green" title="E147">A206</font></span></a>, <a class="txt" href="#E162:67"><span class="lab"><font color="Green" title="E145">A204</font></span></a>, <a class="txt" href="#E79:67"><span class="lab"><font color="Green" title="E96">A111</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E200:67"/><span class="kw">then </span>
 <a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c32">x1</font>,<font color="Maroon" title="c33">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:67"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E16:67"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="txt" href="#E196:67"><span class="lab"><font color="Green" title="E157">A216</font></span></a>, <a class="txt" href="#E127:67"><span class="lab"><font color="Green" title="E130">A189</font></span></a>, <a class="ref" href="weierstr.html#T40" title="WEIERSTR:th.40">WEIERSTR:40</a></span>;<br/>
<a NAME="E201:67"/><span class="kw">then </span>
 <a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">p11</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c31">q11</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jgraph_1.html#T45" target="_self" title="JGRAPH_1:th.45">Th45</a></span>;<br/>
<a NAME="E202:67"/><span class="kw">then </span>
 <a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E194:67"><span class="lab"><font color="Green" title="E156">A215</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E203:67"/><span class="kw">then </span>
<span class="p1">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">5 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/>
<a NAME="E204:67"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 5</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c8">P9</font>,<font color="Maroon" title="c9">Q9</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 5</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:67"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></span>;<br/>
<a NAME="E205:67"/><span class="kw">then </span>
4 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 5 <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="#E31:67"><span class="lab"><font color="Green" title="E62">A20</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E206:67"/>
contradiction
 ; <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>
