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<span class="kw">reconsider </span><font color="Maroon" title="c1">a</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c2">b</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="c3">T</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T</font><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><br/> for <font color="Olive" title="b4">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">T</font><br/> for <font color="Olive" title="b5">S</font> being 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>   st <font color="Olive" title="b5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Olive" title="b4">p</font>,<font color="Olive" title="b3">G</font> holds <br/>for <font color="Olive" title="b6">e1</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>  st <font color="Olive" title="b6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/>(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <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="Olive" title="b5">S</font> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Olive" title="b5">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">e1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T</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="c4">A</font>,<font color="Maroon" title="c5">B</font><br/> for <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">T</font><br/> for <font color="Olive" title="b3">S</font> being 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>   st <font color="Olive" title="b3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b1">G</font> holds <br/>for <font color="Olive" title="b4">e1</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>  st <font color="Olive" title="b4">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/>(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <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="Olive" title="b3">S</font> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Olive" title="b3">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">e1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">G</font> be    <a href="urysohn3.html#M2" title="URYSOHN3:mode.2">Rain</a> of <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">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">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">T</font><br/> for <font color="Olive" title="b2">S</font> being 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>   st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c6">G</font> holds <br/>for <font color="Olive" title="b3">e1</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>  st <font color="Olive" title="b3">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/>(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <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="Olive" title="b2">S</font> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Olive" title="b2">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">e1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">T</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">S</font> being 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>   st <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">G</font> holds <br/>for <font color="Olive" title="b2">e1</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>  st <font color="Olive" title="b2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/>(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <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="Olive" title="b1">S</font> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Olive" title="b1">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">e1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">S</font> be  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> ; <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="c8">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">G</font> implies for <font color="Olive" title="b1">e1</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>  st <font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/>(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <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> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c8">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e1</font> ) )</span><br/>









<span class="kw">consider </span><font color="Maroon" title="c9">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:26"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">s</font> = <font color="Maroon" title="c9">s</font> 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="txt" href="#E3:26"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:26"/>
<font color="Maroon" title="c8">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">G</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">e1</font> being  <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>  st <font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/>(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <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> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c8">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e1</font> )</span><br/>

<a NAME="E6:26"/><span class="kw">then </span>
<font color="Maroon" title="c8">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#T15" target="_self" title="URYSOHN3:th.15">Th15</a></span>;<br/>
<a NAME="E7:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c8">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn2.html#T32" title="URYSOHN2:th.32">URYSOHN2:32</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c11">e1</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"> ( <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <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> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c8">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">e1</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E8:26"/>
<font color="Maroon" title="c11">e1</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"> (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <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> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c8">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">e1</font> )</span><br/>

<a NAME="E9:26"/><span class="kw">then </span>
for <font color="Olive" title="b1">x</font> being  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> holds <br/><font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">e1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E10:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c11">e1</font> is    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of <font color="Maroon" title="c8">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D1" title="XXREAL_2:def.1">XXREAL_2:def 1</a></span>;<br/>
<a NAME="E11:26"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:26"/>
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <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> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c8">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">e1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="xxreal_2.html#T4" title="XXREAL_2:th.4">XXREAL_2:4</a>, <a class="ref" href="xxreal_2.html#D3" title="XXREAL_2:def.3">XXREAL_2:def 3</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/>


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