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<a NAME="E1:30"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="urysohn1.html#T6" title="URYSOHN1:th.6">URYSOHN1:6</a></span>;<br/>
<a NAME="E2:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A2</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#D4" title="URYSOHN1:def.4">URYSOHN1:def 4</a></span>;<br/>
<a NAME="E3:30"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E4:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A3</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c1">e1</font> = 1 as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c2">T</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#V10" title="PRE_TOPC:attr.10">normal</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> holds <br/>for <font color="Olive" title="b3">G</font> being   <a href="urysohn3.html#M2" title="URYSOHN3:mode.2">Rain</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> holds <br/> (  <a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b3">G</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; ( for <font color="Olive" title="b4">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font> holds <br/> (  <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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b3">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b3">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b3">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b3">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</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">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font> implies for <font color="Olive" title="b1">G</font> being   <a href="urysohn3.html#M2" title="URYSOHN3:mode.2">Rain</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> holds <br/> (  <a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b1">G</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; ( for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font> holds <br/> (  <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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E6:30"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <a href="urysohn3.html#M2" title="URYSOHN3:mode.2">Rain</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> holds <br/> (  <a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b1">G</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; ( for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font> holds <br/> (  <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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">G</font> be    <a href="urysohn3.html#M2" title="URYSOHN3:mode.2">Rain</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font> holds <br/> (  <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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) ) )</span><br/>

<a NAME="E7:30"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="urysohn1.html#T6" title="URYSOHN1:th.6">URYSOHN1:6</a></span>;<br/>
<a NAME="E8:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A6</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#D4" title="URYSOHN1:def.4">URYSOHN1:def 4</a></span>;<br/>
<a NAME="E9:30"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E10:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A7</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E11:30"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font> holds <br/> (  <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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_1_1"/>
(  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:30_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
 <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/><div class="add"><a NAME="E2:30_1_1_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:30_1_1_1_1_1"/>
<font color="Maroon" title="c5">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></span>;<br/>
<a NAME="E2:30_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A10</font></span>: 
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:30_1_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</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"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E4:30_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_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="c5">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <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="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></span>;<br/>
<a NAME="E5:30_1_1_1_1_1"/><span class="kw">then </span>
for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E2:30_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:30_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a>, <a class="ref" href="urysohn2.html#T31" title="URYSOHN2:th.31">URYSOHN2:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:30_1_1_1_1"/>
(  <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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="urysohn3.html#D6" target="_self" title="URYSOHN3:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:30_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
 <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/><div class="add"><a NAME="E2:30_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1_1_1_2_1"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</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"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:30_1_1_1_2_1"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
for <font color="Olive" title="b1">s</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>  st  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> not <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">s</font> be   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">s</font> implies  not <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1_1_1_2_1_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">s</font>
 ; <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="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">G</font></span><br/>

<span class="kw">assume </span><a NAME="E2:30_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E40">A16</font></span>: 
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">G</font>
 ; <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="E3:30_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A17</font></span>: 
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c7">s</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:30_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E42">A18</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="urysohn1.html#D4" title="URYSOHN1:def.4">URYSOHN1:def 4</a></span>;<br/>
<a NAME="E7:30_1_1_1_2_1_1"/>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E8:30_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E9:30_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A19</font></span>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E3:30_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E6:30_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">r1</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c11">r2</font> = <font color="Maroon" title="c8">s</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c9">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="ref" href="urysohn1.html#T12" title="URYSOHN1:th.12">URYSOHN1:12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">D</font> = <font color="Maroon" title="c5">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Maroon" title="c9">n</font> as    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c9">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">s</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="ref" href="urysohn1.html#T5" title="URYSOHN1:th.5">URYSOHN1:5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A20</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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/><div class="add"><a NAME="E2:30_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:30_1_1_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E2:30_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E9:30_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E1:30_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A21</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:30_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A22</font></span>: 
<font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c10">r1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c10">r1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></span>;<br/><a NAME="E3:30_1_1_1_2_1_1_1_2"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c10">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c11">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E1:30_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></span>;<br/><a NAME="E4:30_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A23</font></span>: 
<font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c10">r1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c11">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E45">A22</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E5:30_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></span>;<br/><a NAME="E6:30_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A23</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:30_1_1_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E2:30_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E9:30_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:30_1_1_1_2_1"/>
<font color="Maroon" title="c5">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></span>;<br/>
<a NAME="E4:30_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A24</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></span>;<br/>
<a NAME="E5:30_1_1_1_2_1"/><span class="lab"><font color="Green" title="E41">A25</font></span>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E10:30"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:30_1_1_1_2_1"/><span class="lab"><font color="Green" title="E42">A26</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1_2"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E8:30_1_1_1_2_1"/><span class="lab"><font color="Green" title="E43">A27</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_1_1_2_1"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c7">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E10:30_1_1_1_2_1"/><span class="lab"><font color="Green" title="E44">A28</font></span>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Olive" title="b1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_1_1_2_1"><span class="lab"><font color="Green" title="E43">A27</font></span></a>, <a class="ref" href="urysohn1.html#D4" title="URYSOHN1:def.4">URYSOHN1:def 4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_1_1_2_1_2"/>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">a</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_1_1_2_1"><span class="lab"><font color="Green" title="E44">A28</font></span></a>, <a class="ref" href="urysohn1.html#T5" title="URYSOHN1:th.5">URYSOHN1:5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:30_1_1_1_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E5:30_1_1_1_2_1"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="txt" href="#E4:30_1_1_1_2_1"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="txt" href="#E7:30_1_1_1_2_1"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_2_1_2_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">a</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"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:30_1_1_1_2_1_2_2"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1_1_2_1"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="txt" href="#E7:30_1_1_1_2_1"><span class="lab"><font color="Green" title="E42">A26</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">reconsider </span><font color="Maroon" title="c7">S</font> =  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">G</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1_2"><span class="lab"><font color="Green" title="E36">A12</font></span></a></span>;<br/><a NAME="E4:30_1_1_1_2"/><span class="lab"><font color="Green" title="E38">A29</font></span>: 
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn3.html#T17" target="_self" title="URYSOHN3:th.17">Th17</a></span>;<br/><a NAME="E5:30_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A30</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:30_1_1_1_2_2"/>
<font color="Maroon" title="c5">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></span>;<br/>
<a NAME="E2:30_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A31</font></span>: 
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:30_1_1_1_2_2"/><span class="lab"><font color="Green" title="E41">A32</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</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"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E4:30_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_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="c5">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <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="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></span>;<br/>
<a NAME="E5:30_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A33</font></span>: 
for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1_1_2_2"><span class="lab"><font color="Green" title="E41">A32</font></span></a>, <a class="txt" href="#E2:30_1_1_1_2_2"><span class="lab"><font color="Green" title="E40">A31</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">S</font> =  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">G</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a>, <a class="ref" href="urysohn2.html#T31" title="URYSOHN2:th.31">URYSOHN2:31</a></span>;<br/>
<a NAME="E7:30_1_1_1_2_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_1_1_2_2"><span class="lab"><font color="Green" title="E42">A33</font></span></a>, <a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a>, <a class="ref" href="urysohn2.html#T31" title="URYSOHN2:th.31">URYSOHN2:31</a></span>;<br/>
<a NAME="E8:30_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A34</font></span>: 
<font color="Maroon" title="c1">e1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c8">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T4" title="XXREAL_2:th.4">XXREAL_2:4</a></span>;<br/>
<a NAME="E9:30_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c8">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn3.html#D6" target="_self" title="URYSOHN3:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:30_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1_1_2"><span class="lab"><font color="Green" title="E38">A29</font></span></a>, <a class="txt" href="#E8:30_1_1_1_2_2"><span class="lab"><font color="Green" title="E43">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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="E6:30_1_1_1_2"/>
( <font color="Maroon" title="c1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn3.html#D6" target="_self" title="URYSOHN3:def.6">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E7:30_1_1_1_2"/>
(  <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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1_1_2"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E5:30_1_1_1_2"><span class="lab"><font color="Green" title="E39">A30</font></span></a>, <a class="ref" href="urysohn3.html#T17" target="_self" title="URYSOHN3:th.17">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:30_1"/>
(  <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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 ; <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="E12:30"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font> holds   <a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">p</font></span><br/>

<a NAME="E1:30_2"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>   st <font color="Olive" title="b1">Q</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/> ex <font color="Olive" title="b2">H</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b2">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">Q</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font> implies  ex <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Q</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_2_1"/><span class="lab"><font color="Green" title="E36">A35</font></span>: 
<font color="Maroon" title="c7">Q</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E2:30_2_1"/><span class="lab"><font color="Green" title="E37">A36</font></span>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Q</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_2_1"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">Q</font> = <font color="Maroon" title="c7">Q</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/>
<a NAME="E5:30_2_1"/>
( the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  &amp; <font color="Maroon" title="c9">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2_1"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E7:30_2_1"/><span class="lab"><font color="Green" title="E38">A37</font></span>: 
<font color="Maroon" title="c10">r</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">and </span><br/><a NAME="E8:30_2_1"/><span class="lab"><font color="Green" title="E39">A38</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_2_1"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="pcomps_1.html#D5" title="PCOMPS_1:def.5">PCOMPS_1:def 5</a></span>;<br/>
<a NAME="E9:30_2_1"/>
 ex <font color="Olive" title="b1">r0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r0</font> &amp; <font color="Olive" title="b1">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_2_1_1_1"/>
( <font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_2_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A39</font></span>: 
<font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r0</font> &amp; <font color="Olive" title="b1">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c11">r0</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E3:30_2_1_1_1_1"/><span class="lab"><font color="Green" title="E41">A40</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r0</font> &amp; <font color="Maroon" title="c11">r0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_2_1"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">r0</font> = <font color="Maroon" title="c11">r0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c12">r0</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="c12">r0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r0</font> &amp; <font color="Maroon" title="c12">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )</span><br/><span class="kw">thus </span><a NAME="E5:30_2_1_1_1_1"/>
( <font color="Maroon" title="c12">r0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r0</font> &amp; <font color="Maroon" title="c12">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2_1_1_1_1"><span class="lab"><font color="Green" title="E40">A39</font></span></a>, <a class="txt" href="#E3:30_2_1_1_1_1"><span class="lab"><font color="Green" title="E41">A40</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_2_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r0</font> &amp; <font color="Olive" title="b1">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:30_2_1_1_1_2"/>
 ex <font color="Olive" title="b1">r0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r0</font> &amp; <font color="Olive" title="b1">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">r0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:30_2_1"/><span class="lab"><font color="Green" title="E40">A41</font></span>: 
<font color="Maroon" title="c11">r0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font>
 <span class="kw">and </span><br/><a NAME="E12:30_2_1"/><span class="lab"><font color="Green" title="E41">A42</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r0</font> &amp; <font color="Maroon" title="c11">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E14:30_2_1"/><span class="lab"><font color="Green" title="E42">A43</font></span>: 
<font color="Maroon" title="c12">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 <span class="kw">and </span><br/><a NAME="E15:30_2_1"/><span class="lab"><font color="Green" title="E43">A44</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r1</font>
 <span class="kw">and </span><br/><a NAME="E16:30_2_1"/><span class="lab"><font color="Green" title="E44">A45</font></span>: 
<font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">r0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30_2_1"><span class="lab"><font color="Green" title="E41">A42</font></span></a>, <a class="ref" href="urysohn2.html#T28" title="URYSOHN2:th.28">URYSOHN2:28</a></span>;<br/>
<a NAME="E17:30_2_1"/><span class="lab"><font color="Green" title="E45">A46</font></span>: 
<font color="Maroon" title="c12">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E19:30_2_1"/><span class="lab"><font color="Green" title="E46">A47</font></span>: 
<font color="Maroon" title="c12">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c13">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="ref" href="urysohn1.html#D4" title="URYSOHN1:def.4">URYSOHN1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">D</font> = <font color="Maroon" title="c5">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Maroon" title="c13">n</font> as    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c13">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></span>;<br/>
<a NAME="E21:30_2_1"/>
<font color="Maroon" title="c12">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E22:30_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A48</font></span>: 
<font color="Maroon" title="c12">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E23:30_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A49</font></span>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E14:30_2_1"><span class="lab"><font color="Green" title="E42">A43</font></span></a>, <a class="txt" href="#E19:30_2_1"><span class="lab"><font color="Green" title="E46">A47</font></span></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></span>;<br/>
<a NAME="E24:30_2_1"/><span class="lab"><font color="Green" title="E49">A50</font></span>: 
<font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_2_1"><span class="lab"><font color="Green" title="E40">A41</font></span></a>, <a class="txt" href="#E16:30_2_1"><span class="lab"><font color="Green" title="E44">A45</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E25:30_2_1"/><span class="lab"><font color="Green" title="E50">A51</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 <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">r1</font> holds <br/><span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</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="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">r1</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_2_1_2"/>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">r1</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"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font></span><br/>

<a NAME="E2:30_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E15:30_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E17:30_2_1"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="ref" href="urysohn3.html#T19" target="_self" title="URYSOHN3:th.19">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:30_2_1_2"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:30_2_1"><span class="lab"><font color="Green" title="E49">A50</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="E26:30_2_1"/><span class="lab"><font color="Green" title="E51">A52</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">r1</font> = <font color="Maroon" title="c12">r1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c13">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:30_2_1"><span class="lab"><font color="Green" title="E46">A47</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">H</font> = <font color="Maroon" title="c14">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c15">r1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> ;<br/>
<a NAME="E29:30_2_1"/><span class="lab"><font color="Green" title="E52">A53</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c13">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#T12" title="URYSOHN1:th.12">URYSOHN1:12</a></span>;<br/>
<a NAME="E30:30_2_1"/>
 ex <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_2_1_3_1"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_2_1_3_1_1"/><span class="lab"><font color="Green" title="E53">A54</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q</font> )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c16">H</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="c16">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c16">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q</font> )</span><br/><a NAME="E2:30_2_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c16">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_3_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c17">p</font> = <font color="Maroon" title="c8">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c18">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="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c16">H</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:30_2_1_3_1_1_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c16">H</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="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:30_2_1_3_1_1_1"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:30_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E54">A55</font></span>: 
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">H</font>
 <span class="kw">and </span><br/><a NAME="E6:30_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E55">A56</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x1</font>
 <span class="kw">by</span> <span class="lab"><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="c20">y</font> = <font color="Maroon" title="c18">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:30_2_1"><span class="lab"><font color="Green" title="E51">A52</font></span></a>, <a class="txt" href="#E5:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E54">A55</font></span></a>, <a class="txt" href="#E6:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A56</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">q</font> = <font color="Maroon" title="c20">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></span>;<br/>
<a NAME="E9:30_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E56">A57</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:30_2_1"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="txt" href="#E23:30_2_1"><span class="lab"><font color="Green" title="E48">A49</font></span></a>, <a class="txt" href="#E1:30_2_1_3_1_1"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="txt" href="#E5:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E54">A55</font></span></a>, <a class="txt" href="#E6:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A56</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">x1</font> = <font color="Maroon" title="c19">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E54">A55</font></span></a></span>;<br/>
<a NAME="E11:30_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E57">A58</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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30"><span class="lab"><font color="Green" title="E35">A8</font></span></a></span>;<br/>
<a NAME="E12:30_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E58">A59</font></span>: 
<font color="Maroon" title="c21">q</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:30_2_1"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="txt" href="#E23:30_2_1"><span class="lab"><font color="Green" title="E48">A49</font></span></a>, <a class="txt" href="#E1:30_2_1_3_1_1"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="txt" href="#E5:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E54">A55</font></span></a>, <a class="txt" href="#E6:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A56</font></span></a></span>;<br/>
<a NAME="E13:30_2_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A60</font></span>: 
<font color="Maroon" title="c17">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c21">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2_1_3_1_1"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="txt" href="#E6:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A56</font></span></a>, <a class="txt" href="#E11:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E57">A58</font></span></a>, <a class="ref" href="xreal_1.html#T16" title="XREAL_1:th.16">XREAL_1:16</a></span>;<br/>
<a NAME="E14:30_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E60">A61</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E59">A60</font></span></a>, <a class="txt" href="#E9:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E15:30_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E61">A63</font></span>: 
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c20">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font> implies <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></span>;<br/>
<a NAME="E16:30_2_1_3_1_1_1"/>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c21">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c21">q</font>
 
;<br/>
<a NAME="E17:30_2_1_3_1_1_1"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c21">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A57</font></span></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></span>;<br/>
<a NAME="E18:30_2_1_3_1_1_1"/><span class="kw">then </span>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c21">q</font></span>)</span> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2_1_3_1_1"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="txt" href="#E6:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A56</font></span></a>, <a class="txt" href="#E11:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E57">A58</font></span></a>, <a class="txt" href="#E12:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E58">A59</font></span></a>, <a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a>, <a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:30_2_1_3_1_1_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_2_1"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="txt" href="#E15:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E61">A63</font></span></a>, <a class="txt" href="#E14:30_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E60">A61</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:30_2_1_3_1_1"/>
( <font color="Maroon" title="c16">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c16">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E15:30_2_1"><span class="lab"><font color="Green" title="E43">A44</font></span></a>, <a class="txt" href="#E22:30_2_1"><span class="lab"><font color="Green" title="E47">A48</font></span></a>, <a class="txt" href="#E23:30_2_1"><span class="lab"><font color="Green" title="E48">A49</font></span></a>, <a class="txt" href="#E29:30_2_1"><span class="lab"><font color="Green" title="E52">A53</font></span></a>, <a class="txt" href="#E1:30_2_1_3_1_1"><span class="lab"><font color="Green" title="E53">A54</font></span></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a>, <a class="ref" href="urysohn3.html#T18" target="_self" title="URYSOHN3:th.18">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_2_1_3_1_2"/><span class="lab"><font color="Green" title="E53">A64</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q</font> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c8">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E3:30_2_1_3_1_2"/>
 <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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30"><span class="lab"><font color="Green" title="E35">A8</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">r1</font>, <font color="Maroon" title="c19">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E5:30_2_1_3_1_2"/><span class="lab"><font color="Green" title="E54">A65</font></span>: 
<font color="Maroon" title="c18">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:30_2_1_3_1_2"/>
<font color="Maroon" title="c19">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:30_2_1_3_1_2"/><span class="lab"><font color="Green" title="E55">A66</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">r1</font>
 <span class="kw">and </span><br/><a NAME="E8:30_2_1_3_1_2"/><span class="lab"><font color="Green" title="E56">A67</font></span>: 
<font color="Maroon" title="c18">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">and </span><br/><a NAME="E9:30_2_1_3_1_2"/><span class="lab"><font color="Green" title="E57">A68</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">r2</font>
 <span class="kw">and </span><br/><a NAME="E10:30_2_1_3_1_2"/><span class="lab"><font color="Green" title="E58">A69</font></span>: 
<font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_2_1"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="txt" href="#E1:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E53">A64</font></span></a>, <a class="ref" href="urysohn2.html#T35" title="URYSOHN2:th.35">URYSOHN2:35</a></span>;<br/><a NAME="E11:30_2_1_3_1_2"/>
( <font color="Maroon" title="c18">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  or <font color="Maroon" title="c18">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E54">A65</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E12:30_2_1_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c18">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  or <font color="Maroon" title="c18">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E13:30_2_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A70</font></span>: 
<font color="Maroon" title="c18">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">B</font> = <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">r1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">C</font> = <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c20">B</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c22">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E17:30_2_1_3_1_2"/><span class="lab"><font color="Green" title="E60">A71</font></span>: 
<font color="Maroon" title="c22">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 <span class="kw">and </span><br/><a NAME="E18:30_2_1_3_1_2"/><span class="lab"><font color="Green" title="E61">A72</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">r3</font>
 <span class="kw">and </span><br/><a NAME="E19:30_2_1_3_1_2"/><span class="lab"><font color="Green" title="E62">A73</font></span>: 
<font color="Maroon" title="c22">r3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E57">A68</font></span></a>, <a class="ref" href="urysohn2.html#T29" title="URYSOHN2:th.29">URYSOHN2:29</a></span>;<br/><a NAME="E20:30_2_1_3_1_2"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c20">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c20">B</font>
 ;<br/><a NAME="E21:30_2_1_3_1_2"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c20">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></span>;<br/><a NAME="E22:30_2_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A74</font></span>: 
<font color="Maroon" title="c21">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">A</font> = <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">r3</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E60">A71</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c24">H</font> = <font color="Maroon" title="c23">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">C</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="c24">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c24">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q</font> )</span><br/><a NAME="E24:30_2_1_3_1_2"/><span class="lab"><font color="Green" title="E64">A75</font></span>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c24">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_3_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c25">x</font> = <font color="Maroon" title="c17">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  ;<br/>
<span class="kw">let </span><font color="Maroon" title="c26">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="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c24">H</font> or <font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c27">p</font> = <font color="Maroon" title="c25">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">assume </span><a NAME="E3:30_2_1_3_1_2_1"/>
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c24">H</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="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:30_2_1_3_1_2_1"/>
<font color="Maroon" title="c28">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:30_2_1_3_1_2_1"/><span class="lab"><font color="Green" title="E65">A76</font></span>: 
<font color="Maroon" title="c28">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">H</font>
 <span class="kw">and </span><br/><a NAME="E7:30_2_1_3_1_2_1"/><span class="lab"><font color="Green" title="E66">A77</font></span>: 
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x1</font>
 <span class="kw">by</span> <span class="lab"><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">x1</font> = <font color="Maroon" title="c28">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E65">A76</font></span></a></span>;<br/>
<a NAME="E9:30_2_1_3_1_2_1"/><span class="lab"><font color="Green" title="E67">A78</font></span>: 
<font color="Maroon" title="c29">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">r3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E65">A76</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c30">y</font> = <font color="Maroon" title="c26">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E66">A77</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c31">y</font> = <font color="Maroon" title="c30">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:30_2_1"><span class="lab"><font color="Green" title="E51">A52</font></span></a>, <a class="txt" href="#E6:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E65">A76</font></span></a>, <a class="txt" href="#E7:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E66">A77</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c32">q</font> = <font color="Maroon" title="c31">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E13:30_2_1_3_1_2_1"/><span class="lab"><font color="Green" title="E68">A79</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font>
 
;<br/>
<a NAME="E14:30_2_1_3_1_2_1"/><span class="lab"><font color="Green" title="E69">A80</font></span>: 
not <font color="Maroon" title="c22">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E18:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E61">A72</font></span></a></span>;<br/>
<a NAME="E15:30_2_1_3_1_2_1"/>
( <font color="Maroon" title="c22">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  or <font color="Maroon" title="c22">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E60">A71</font></span></a>, <a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E16:30_2_1_3_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c22">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c22">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  or <font color="Maroon" title="c22">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E17:30_2_1_3_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E69">A80</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a>, <a class="ref" href="xxreal_1.html#T233" title="XXREAL_1:th.233">XXREAL_1:233</a></span>;<br/>
<a NAME="E18:30_2_1_3_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">r3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E9:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E67">A78</font></span></a>, <a class="txt" href="#E14:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E69">A80</font></span></a>, <a class="ref" href="urysohn3.html#T19" target="_self" title="URYSOHN3:th.19">Th19</a></span>;<br/>
<a NAME="E19:30_2_1_3_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">r2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E62">A73</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E20:30_2_1_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A81</font></span>: 
<font color="Maroon" title="c32">q</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c27">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E56">A67</font></span></a>, <a class="txt" href="#E7:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E66">A77</font></span></a>, <a class="ref" href="xreal_1.html#T16" title="XREAL_1:th.16">XREAL_1:16</a></span>;<br/>
<a NAME="E21:30_2_1_3_1_2_1"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E58">A69</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E22:30_2_1_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A82</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E68">A79</font></span></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></span>;<br/>
<a NAME="E23:30_2_1_3_1_2_1"/><span class="lab"><font color="Green" title="E72">A83</font></span>: 
<font color="Maroon" title="c29">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c20">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E65">A76</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E24:30_2_1_3_1_2_1"/>
not <font color="Maroon" title="c29">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_3_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:30_2_1_3_1_2_1_1"/><span class="lab"><font color="Green" title="E73">A84</font></span>: 
<font color="Maroon" title="c20">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c20">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:30_2_1_3_1_2_1_1"/>
<font color="Maroon" title="c29">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E3:30_2_1_3_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E72">A83</font></span></a>, <a class="txt" href="#E1:30_2_1_3_1_2_1_1"><span class="lab"><font color="Green" title="E73">A84</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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="E25:30_2_1_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A85</font></span>: 
<font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E9:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E57">A68</font></span></a>, <a class="txt" href="#E13:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E59">A70</font></span></a>, <a class="txt" href="#E7:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E66">A77</font></span></a>, <a class="ref" href="urysohn3.html#T18" target="_self" title="URYSOHN3:th.18">Th18</a>, <a class="ref" href="xreal_1.html#T16" title="XREAL_1:th.16">XREAL_1:16</a></span>;<br/>
<a NAME="E26:30_2_1_3_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E58">A69</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E27:30_2_1_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A86</font></span>: 
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c31">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font></span>)</span> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E71">A82</font></span></a>, <a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a>, <a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></span>;<br/>
<a NAME="E28:30_2_1_3_1_2_1"/><span class="lab"><font color="Green" title="E75">A87</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_3_1_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_2_1_3_1_2_1_2"/><span class="lab"><font color="Green" title="E76">A88</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_3_1_2_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_2_1_3_1_2_1_2_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font> or <font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_3_1_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_2_1_3_1_2_1_2_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:30_2_1_3_1_2_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E58">A69</font></span></a>, <a class="txt" href="#E25:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E73">A85</font></span></a>, <a class="txt" href="#E1:30_2_1_3_1_2_1_2"><span class="lab"><font color="Green" title="E76">A88</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_3_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_2_1_3_1_2_1_2_1_2"/>
<font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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/><div class="add"><a NAME="E2:30_2_1_3_1_2_1_2_1_2"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:30_2_1_3_1_2_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E58">A69</font></span></a>, <a class="txt" href="#E20:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E70">A81</font></span></a>, <a class="txt" href="#E1:30_2_1_3_1_2_1_2"><span class="lab"><font color="Green" title="E76">A88</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E29:30_2_1_3_1_2_1"/>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c31">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font> implies <font color="Maroon" title="c31">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c10">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:30_2_1_3_1_2_1"/>
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_2_1"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="txt" href="#E27:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E74">A86</font></span></a>, <a class="txt" href="#E28:30_2_1_3_1_2_1"><span class="lab"><font color="Green" title="E75">A87</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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="E25:30_2_1_3_1_2"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30"><span class="lab"><font color="Green" title="E35">A8</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">r4</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E27:30_2_1_3_1_2"/><span class="lab"><font color="Green" title="E65">A89</font></span>: 
<font color="Maroon" title="c25">r4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 <span class="kw">and </span><br/><a NAME="E28:30_2_1_3_1_2"/><span class="lab"><font color="Green" title="E66">A90</font></span>: 
<font color="Maroon" title="c18">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">r4</font>
 <span class="kw">and </span><br/><a NAME="E29:30_2_1_3_1_2"/><span class="lab"><font color="Green" title="E67">A91</font></span>: 
<font color="Maroon" title="c25">r4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E55">A66</font></span></a>, <a class="txt" href="#E8:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E56">A67</font></span></a>, <a class="ref" href="urysohn2.html#T28" title="URYSOHN2:th.28">URYSOHN2:28</a></span>;<br/><a NAME="E30:30_2_1_3_1_2"/><span class="lab"><font color="Green" title="E68">A92</font></span>: 
<font color="Maroon" title="c25">r4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E65">A89</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E31:30_2_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">r4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E32:30_2_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A93</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c20">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E28:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E66">A90</font></span></a>, <a class="txt" href="#E13:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E59">A70</font></span></a>, <a class="ref" href="urysohn3.html#T13" target="_self" title="URYSOHN3:th.13">Th13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">r4</font> = <font color="Maroon" title="c25">r4</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E68">A92</font></span></a>, <a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E34:30_2_1_3_1_2"/>
not <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c26">r4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E7:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E55">A66</font></span></a>, <a class="txt" href="#E28:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E66">A90</font></span></a>, <a class="txt" href="#E29:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E67">A91</font></span></a>, <a class="ref" href="urysohn3.html#T20" target="_self" title="URYSOHN3:th.20">Th20</a></span>;<br/><a NAME="E35:30_2_1_3_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c20">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E69">A93</font></span></a></span>;<br/><a NAME="E36:30_2_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A94</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E37:30_2_1_3_1_2"/>
( <font color="Maroon" title="c23">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">r3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E17:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E60">A71</font></span></a>, <a class="txt" href="#E18:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E61">A72</font></span></a>, <a class="ref" href="urysohn3.html#T12" target="_self" title="URYSOHN3:th.12">Th12</a>, <a class="ref" href="urysohn3.html#T18" target="_self" title="URYSOHN3:th.18">Th18</a></span>;<br/><span class="kw">hence </span><a NAME="E38:30_2_1_3_1_2"/>
( <font color="Maroon" title="c24">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c24">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E63">A74</font></span></a>, <a class="txt" href="#E36:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E70">A94</font></span></a>, <a class="txt" href="#E24:30_2_1_3_1_2"><span class="lab"><font color="Green" title="E64">A75</font></span></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E31:30_2_1"/>
 ex <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Olive" title="b1">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:30_2"/>
 <a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T48" title="TMAP_1:th.48">TMAP_1:48</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:30"/>
(  <a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font> holds <br/> (  <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="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="ref" href="tmap_1.html#T49" title="TMAP_1:th.49">TMAP_1:49</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>
