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<span class="kw">let </span><font color="Maroon" title="c1">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"> ( <font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V10" title="PRE_TOPC:attr.10">normal</a>  implies 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="c1">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>  holds <br/>for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Olive" title="b3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>  st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b5">r1</font>, <font color="Olive" title="b6">r2</font> being   <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="Olive" title="b3">n</font>  st <font color="Olive" title="b5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">r2</font> holds <br/>( <font color="Olive" title="b4">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b5">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b4">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b6">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b5">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b6">r2</font> ) ) holds <br/> ex <font color="Olive" title="b5">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> st <br/>( <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b6">r1</font>, <font color="Olive" title="b7">r2</font>, <font color="Olive" title="b8">r</font> being   <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> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b6">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b7">r2</font> holds <br/>( <font color="Olive" title="b5">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b6">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b5">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b7">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b6">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b7">r2</font> &amp; ( <font color="Olive" title="b8">r</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="b3">n</font> implies <font color="Olive" title="b5">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">r</font> ) ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:45"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="pre_topc.html#V10" title="PRE_TOPC:attr.10">normal</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="c1">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>  holds <br/>for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Olive" title="b3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>  st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b5">r1</font>, <font color="Olive" title="b6">r2</font> being   <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="Olive" title="b3">n</font>  st <font color="Olive" title="b5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">r2</font> holds <br/>( <font color="Olive" title="b4">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b5">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b4">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b6">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b5">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b6">r2</font> ) ) holds <br/> ex <font color="Olive" title="b5">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> st <br/>( <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b6">r1</font>, <font color="Olive" title="b7">r2</font>, <font color="Olive" title="b8">r</font> being   <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> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b6">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b7">r2</font> holds <br/>( <font color="Olive" title="b5">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b6">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b5">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b7">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b6">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b7">r2</font> &amp; ( <font color="Olive" title="b8">r</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="b3">n</font> implies <font color="Olive" title="b5">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">r</font> ) ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">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="c1">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="c2">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>  implies for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>  st <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</font> being   <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="Olive" title="b1">n</font>  st <font color="Olive" title="b3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">r2</font> holds <br/>( <font color="Olive" title="b2">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b2">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b4">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b4">r2</font> ) ) holds <br/> ex <font color="Olive" title="b3">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> st <br/>( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b4">r1</font>, <font color="Olive" title="b5">r2</font>, <font color="Olive" title="b6">r</font> being   <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> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b4">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">r2</font> holds <br/>( <font color="Olive" title="b3">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b4">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b3">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b5">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b4">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b5">r2</font> &amp; ( <font color="Olive" title="b6">r</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> implies <font color="Olive" title="b3">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">r</font> ) ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:45"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c2">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> 
 ; <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">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>  st <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</font> being   <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="Olive" title="b1">n</font>  st <font color="Olive" title="b3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">r2</font> holds <br/>( <font color="Olive" title="b2">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b2">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b4">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b4">r2</font> ) ) holds <br/> ex <font color="Olive" title="b3">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> st <br/>( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b4">r1</font>, <font color="Olive" title="b5">r2</font>, <font color="Olive" title="b6">r</font> being   <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> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b4">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">r2</font> holds <br/>( <font color="Olive" title="b3">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b4">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b3">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b5">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b4">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b5">r2</font> &amp; ( <font color="Olive" title="b6">r</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> implies <font color="Olive" title="b3">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">r</font> ) ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>  st <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being   <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="c4">n</font>  st <font color="Olive" title="b2">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r2</font> holds <br/>( <font color="Olive" title="b1">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b1">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r2</font> ) ) holds <br/> ex <font color="Olive" title="b2">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> st <br/>( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</font>, <font color="Olive" title="b5">r</font> being   <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b3">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">r2</font> holds <br/>( <font color="Olive" title="b2">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b2">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b4">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b4">r2</font> &amp; ( <font color="Olive" title="b5">r</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="c4">n</font> implies <font color="Olive" title="b2">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">r</font> ) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">G</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <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="c4">n</font>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/>( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r2</font> ) ) implies  ex <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> st <br/>( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font>, <font color="Olive" title="b4">r</font> being   <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b2">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r2</font> holds <br/>( <font color="Olive" title="b1">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b1">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r2</font> &amp; ( <font color="Olive" title="b4">r</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="c4">n</font> implies <font color="Olive" title="b1">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">r</font> ) ) ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:45"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E4:45"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c3">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:45"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <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="c4">n</font>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/>( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r2</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">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> st <br/>( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font>, <font color="Olive" title="b4">r</font> being   <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b2">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r2</font> holds <br/>( <font color="Olive" title="b1">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b1">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r2</font> &amp; ( <font color="Olive" title="b4">r</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="c4">n</font> implies <font color="Olive" title="b1">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">r</font> ) ) ) )</span><br/>

<a NAME="E6:45"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
for <font color="Olive" title="b1">r</font> being   <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="c4">n</font>  holds <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">r</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">r</font> be    <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="c4">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c6">r</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_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="E1:45_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> 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="c6">r</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#T5" target="_self" title="URYSOHN1:th.5">Th5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c6">r</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"><span class="kw">hence </span><a NAME="E2:45_1_1_1"/>
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c6">r</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E26">A3</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_1_1_2"/><span class="lab"><font color="Green" title="E30">A7</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="c6">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c6">r</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"><span class="kw">reconsider </span><font color="Maroon" title="c7">q1</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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="c4">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#T12" target="_self" title="URYSOHN1:th.12">Th12</a></span>;<br/><a NAME="E3:45_1_1_2"/>
( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c7">q1</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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c7">q1</font></span>)</span> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c7">q1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c6">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E1:45_1_1_2"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></span>;<br/><a NAME="E4:45_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c7">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c6">r</font>
 <span class="kw">by</span> <span class="lab"><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="E5:45_1_1_2"/>
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c6">r</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E26">A3</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="c6">S</font> = <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font></span>)</span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#T15" target="_self" title="URYSOHN1:th.15">Th15</a></span>;<br/>
<a NAME="E8:45"/><span class="lab"><font color="Green" title="E30">A8</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#T12" target="_self" title="URYSOHN1:th.12">Th12</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">S</font>,  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>] <span class="kw">means </span>for <font color="Olive" title="b1">r</font> being   <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> holds <br/>for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being   <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="c4">n</font>  st <font color="Olive" title="b2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Olive" title="b1">r</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive" title="b3">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Olive" title="b1">r</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> holds <br/>$2 is    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r1</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r2</font>;<br/>
<a NAME="E9:45"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">S</font>  ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">S</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">y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<a NAME="E1:45_2"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
not <font color="Maroon" title="c7">x</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="c4">n</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E3:45_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c8">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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="c4">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c8">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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="c4">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="urysohn1.html#T20" target="_self" title="URYSOHN1:th.20">Th20</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">q1</font>, <font color="Maroon" title="c10">q2</font> being    <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="c4">n</font><span class="kw"> such that </span><br/><a NAME="E5:45_2"/><span class="lab"><font color="Green" title="E33">A11</font></span>: 
( <font color="Maroon" title="c9">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c8">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c8">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">Q0</font> being    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">q1</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">q2</font>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">Q0</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">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c11">Q0</font>]</span><br/>

<span class="kw">thus </span><a NAME="E7:45_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c11">Q0</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_2"><span class="lab"><font color="Green" title="E33">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c7">D</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">S</font>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:45"/><span class="lab"><font color="Green" title="E32">A12</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">S</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c7">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E9:45"><span class="lab"><font color="Green" title="E31">A9</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>] <span class="kw">means </span>for <font color="Olive" title="b1">r</font> being   <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> holds <br/>( ( <font color="Olive" title="b1">r</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="c4">n</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> ) &amp; ( not <font color="Olive" title="b1">r</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="c4">n</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> ) );<br/>
<a NAME="E12:45"/><span class="lab"><font color="Green" title="E33">A13</font></span>: 
for <font color="Olive" title="b1">x</font> being   <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>; <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">y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_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="E1:45_3_1"/>
( <font color="Maroon" title="c8">x</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="c4">n</font> or  not <font color="Maroon" title="c8">x</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="c4">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_3_1_1"/>
<font color="Maroon" title="c8">x</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="c4">n</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">y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</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="c4">n</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c10">y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E4:45_3_1_1"/><span class="lab"><font color="Green" title="E34">A14</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">x</font>
 ;<br/><span class="kw">take </span>
<font color="Maroon" title="c10">y</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">S<sub>2</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">y</font>]</span><br/><span class="kw">thus </span><a NAME="E5:45_3_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_3_1_1"><span class="lab"><font color="Green" title="E34">A14</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_3_1_2"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
not <font color="Maroon" title="c8">x</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="c4">n</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">y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">S</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/><span class="kw">consider </span><font color="Maroon" title="c10">y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E4:45_3_1_2"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
 ;<br/><span class="kw">take </span>
<font color="Maroon" title="c10">y</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">S<sub>2</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">y</font>]</span><br/><span class="kw">thus </span><a NAME="E5:45_3_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_3_1_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E4:45_3_1_2"><span class="lab"><font color="Green" title="E35">A16</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">consider </span><font color="Maroon" title="c8">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E14:45"/><span class="lab"><font color="Green" title="E34">A17</font></span>: 
for <font color="Olive" title="b1">x</font> being   <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E12:45"><span class="lab"><font color="Green" title="E33">A13</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c8">F</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="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">r</font> being   <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/>( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r2</font> &amp; ( <font color="Olive" title="b3">r</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="c4">n</font> implies <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">r</font> ) ) ) )</span><br/>

<a NAME="E15:45"/>
(  <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="c4">n</font> &amp; 1 <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="c4">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#T12" target="_self" title="URYSOHN1:th.12">Th12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:45"/>
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">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="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <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="#E3:45"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E30">A8</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"> for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">r</font> being   <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/>( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r2</font> &amp; ( <font color="Olive" title="b3">r</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="c4">n</font> implies <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">r</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">r1</font>, <font color="Maroon" title="c10">r2</font>, <font color="Maroon" title="c11">r</font> be    <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r2</font> implies ( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> &amp; ( <font color="Maroon" title="c11">r</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="c4">n</font> implies <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">r</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E17:45"/><span class="lab"><font color="Green" title="E35">A18</font></span>: 
<font color="Maroon" title="c9">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r2</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="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> &amp; ( <font color="Maroon" title="c11">r</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="c4">n</font> implies <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">r</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E18:45"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</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="c11">r</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="c4">n</font> implies <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">r</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_4"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_4_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="E1:45_4_1_1"/>
( ( <font color="Maroon" title="c9">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="c4">n</font> &amp; <font color="Maroon" title="c10">r2</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="c4">n</font> ) or ( <font color="Maroon" title="c9">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="c4">n</font> &amp; not <font color="Maroon" title="c10">r2</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="c4">n</font> ) or ( not <font color="Maroon" title="c9">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="c4">n</font> &amp; <font color="Maroon" title="c10">r2</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="c4">n</font> ) or ( not <font color="Maroon" title="c9">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="c4">n</font> &amp; not <font color="Maroon" title="c10">r2</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="c4">n</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_4_1_1_1"/><span class="lab"><font color="Green" title="E36">A19</font></span>: 
( <font color="Maroon" title="c9">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="c4">n</font> &amp; <font color="Maroon" title="c10">r2</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="c4">n</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="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )</span><br/><div class="add"><a NAME="E2:45_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A20</font></span>: 
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r1</font> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E34">A17</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">r1</font> = <font color="Maroon" title="c9">r1</font>, <font color="Maroon" title="c13">r2</font> = <font color="Maroon" title="c10">r2</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="c4">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_4_1_1_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a></span>;<br/><a NAME="E4:45_4_1_1_1"/>
<font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:45"><span class="lab"><font color="Green" title="E35">A18</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:45_4_1_1_1"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E2:45_4_1_1_1"><span class="lab"><font color="Green" title="E37">A20</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_4_1_1_2"/><span class="lab"><font color="Green" title="E36">A21</font></span>: 
( <font color="Maroon" title="c9">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="c4">n</font> &amp; not <font color="Maroon" title="c10">r2</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="c4">n</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="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )</span><br/><div class="add"><a NAME="E2:45_4_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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="c4">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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="c4">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#T20" target="_self" title="URYSOHN1:th.20">Th20</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">q1</font>, <font color="Maroon" title="c13">q2</font> being    <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="c4">n</font><span class="kw"> such that </span><br/><a NAME="E4:45_4_1_1_2"/><span class="lab"><font color="Green" title="E37">A22</font></span>: 
<font color="Maroon" title="c12">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:45_4_1_1_2"/><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<font color="Maroon" title="c13">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E6:45_4_1_1_2"/><span class="lab"><font color="Green" title="E39">A24</font></span>: 
<font color="Maroon" title="c9">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:45"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E4:45_4_1_1_2"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="urysohn1.html#T22" target="_self" title="URYSOHN1:th.22">Th22</a></span>;<br/><a NAME="E7:45_4_1_1_2"/>
<font color="Maroon" title="c10">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_4_1_1_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E8:45_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A25</font></span>: 
<font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r2</font> is    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E4:45_4_1_1_2"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E5:45_4_1_1_2"><span class="lab"><font color="Green" title="E38">A23</font></span></a></span>;<br/><a NAME="E9:45_4_1_1_2"/><span class="lab"><font color="Green" title="E41">A26</font></span>: 
<font color="Maroon" title="c12">q1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_4_1_1_2"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E5:45_4_1_1_2"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="ref" href="urysohn1.html#T18" target="_self" title="URYSOHN1:th.18">Th18</a></span>;<br/><a NAME="E10:45_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A27</font></span>: 
( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/><a NAME="E11:45_4_1_1_2"/><span class="lab"><font color="Green" title="E43">A28</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E1:45_4_1_1_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a></span>;<br/><a NAME="E12:45_4_1_1_2"/><span class="lab"><font color="Green" title="E44">A29</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E29">A6</font></span></a></span>;<br/><a NAME="E13:45_4_1_1_2"/><span class="lab"><font color="Green" title="E45">A30</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</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="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E9:45_4_1_1_2"><span class="lab"><font color="Green" title="E41">A26</font></span></a></span>;<br/><a NAME="E14:45_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A31</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E11:45_4_1_1_2"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="txt" href="#E8:45_4_1_1_2"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="txt" href="#E12:45_4_1_1_2"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="txt" href="#E10:45_4_1_1_2"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_4_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_4_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_4_1_1_2_1_1"/>
( <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q1</font> or <font color="Maroon" title="c9">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_4_1_1_2"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_4_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_4_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q1</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="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:45_4_1_1_2_1_1_1"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E11:45_4_1_1_2"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="txt" href="#E8:45_4_1_1_2"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="txt" href="#E12:45_4_1_1_2"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="txt" href="#E13:45_4_1_1_2"><span class="lab"><font color="Green" title="E45">A30</font></span></a>, <a class="txt" href="#E10:45_4_1_1_2"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="txt" href="#E14:45_4_1_1_2"><span class="lab"><font color="Green" title="E46">A31</font></span></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</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="45_4_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_4_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E47">A32</font></span>: 
<font color="Maroon" title="c9">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q1</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="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )</span><br/><div class="add"><a NAME="E2:45_4_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E48">A33</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</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/><span class="kw">consider </span><font color="Maroon" title="c14">q0</font> being    <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="c4">n</font><span class="kw"> such that </span><br/><a NAME="E4:45_4_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E49">A34</font></span>: 
<font color="Maroon" title="c14">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_4_1_1_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a></span>;<br/><a NAME="E5:45_4_1_1_2_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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c14">q0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E1:45_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E47">A32</font></span></a>, <a class="txt" href="#E4:45_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E49">A34</font></span></a></span>;<br/><a NAME="E6:45_4_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E4:45_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E49">A34</font></span></a></span>;<br/><a NAME="E7:45_4_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A35</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E48">A33</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E8:45_4_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E51">A36</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c14">q0</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="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E1:45_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E47">A32</font></span></a>, <a class="txt" href="#E4:45_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E49">A34</font></span></a></span>;<br/><a NAME="E9:45_4_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E52">A37</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</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="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E1:45_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E47">A32</font></span></a>, <a class="txt" href="#E4:45_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E49">A34</font></span></a></span>;<br/><a NAME="E10:45_4_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E11:45_4_1_1_2"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="txt" href="#E8:45_4_1_1_2"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="txt" href="#E12:45_4_1_1_2"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="txt" href="#E10:45_4_1_1_2"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E11:45_4_1_1_2_1_1_2"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E11:45_4_1_1_2"><span class="lab"><font color="Green" title="E43">A28</font></span></a>, <a class="txt" href="#E8:45_4_1_1_2"><span class="lab"><font color="Green" title="E40">A25</font></span></a>, <a class="txt" href="#E12:45_4_1_1_2"><span class="lab"><font color="Green" title="E44">A29</font></span></a>, <a class="txt" href="#E10:45_4_1_1_2"><span class="lab"><font color="Green" title="E42">A27</font></span></a>, <a class="txt" href="#E4:45_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="txt" href="#E8:45_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E51">A36</font></span></a>, <a class="txt" href="#E9:45_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E52">A37</font></span></a>, <a class="txt" href="#E7:45_4_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E50">A35</font></span></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E16:45_4_1_1_2"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_4_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:45_4_1_1_3"/><span class="lab"><font color="Green" title="E36">A38</font></span>: 
( not <font color="Maroon" title="c9">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="c4">n</font> &amp; <font color="Maroon" title="c10">r2</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="c4">n</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="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )</span><br/><div class="add"><a NAME="E2:45_4_1_1_3"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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="c4">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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="c4">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#T20" target="_self" title="URYSOHN1:th.20">Th20</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">q1</font>, <font color="Maroon" title="c13">q2</font> being    <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="c4">n</font><span class="kw"> such that </span><br/><a NAME="E4:45_4_1_1_3"/><span class="lab"><font color="Green" title="E37">A39</font></span>: 
<font color="Maroon" title="c12">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:45_4_1_1_3"/><span class="lab"><font color="Green" title="E38">A40</font></span>: 
<font color="Maroon" title="c13">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E6:45_4_1_1_3"/><span class="lab"><font color="Green" title="E39">A41</font></span>: 
<font color="Maroon" title="c13">q2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:45"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E5:45_4_1_1_3"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="ref" href="urysohn1.html#T22" target="_self" title="URYSOHN1:th.22">Th22</a></span>;<br/><a NAME="E7:45_4_1_1_3"/>
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_4_1_1_3"><span class="lab"><font color="Green" title="E36">A38</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E8:45_4_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A42</font></span>: 
<font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r1</font> is    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E4:45_4_1_1_3"><span class="lab"><font color="Green" title="E37">A39</font></span></a>, <a class="txt" href="#E5:45_4_1_1_3"><span class="lab"><font color="Green" title="E38">A40</font></span></a></span>;<br/><a NAME="E9:45_4_1_1_3"/><span class="lab"><font color="Green" title="E41">A43</font></span>: 
<font color="Maroon" title="c12">q1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_4_1_1_3"><span class="lab"><font color="Green" title="E37">A39</font></span></a>, <a class="txt" href="#E5:45_4_1_1_3"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="ref" href="urysohn1.html#T18" target="_self" title="URYSOHN1:th.18">Th18</a></span>;<br/><a NAME="E10:45_4_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A44</font></span>: 
( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/><a NAME="E11:45_4_1_1_3"/><span class="lab"><font color="Green" title="E43">A45</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E1:45_4_1_1_3"><span class="lab"><font color="Green" title="E36">A38</font></span></a></span>;<br/><a NAME="E12:45_4_1_1_3"/><span class="lab"><font color="Green" title="E44">A46</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E29">A6</font></span></a></span>;<br/><a NAME="E13:45_4_1_1_3"/><span class="lab"><font color="Green" title="E45">A47</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</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="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E9:45_4_1_1_3"><span class="lab"><font color="Green" title="E41">A43</font></span></a></span>;<br/><a NAME="E14:45_4_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A48</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E11:45_4_1_1_3"><span class="lab"><font color="Green" title="E43">A45</font></span></a>, <a class="txt" href="#E8:45_4_1_1_3"><span class="lab"><font color="Green" title="E40">A42</font></span></a>, <a class="txt" href="#E12:45_4_1_1_3"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="txt" href="#E10:45_4_1_1_3"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_4_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_4_1_1_3_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="E1:45_4_1_1_3_1_1"/>
( <font color="Maroon" title="c13">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r2</font> or <font color="Maroon" title="c13">q2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_4_1_1_3"><span class="lab"><font color="Green" title="E39">A41</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_4_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_4_1_1_3_1_1_1"/>
<font color="Maroon" title="c13">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r2</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="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:45_4_1_1_3_1_1_1"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E11:45_4_1_1_3"><span class="lab"><font color="Green" title="E43">A45</font></span></a>, <a class="txt" href="#E8:45_4_1_1_3"><span class="lab"><font color="Green" title="E40">A42</font></span></a>, <a class="txt" href="#E12:45_4_1_1_3"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="txt" href="#E13:45_4_1_1_3"><span class="lab"><font color="Green" title="E45">A47</font></span></a>, <a class="txt" href="#E10:45_4_1_1_3"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="txt" href="#E14:45_4_1_1_3"><span class="lab"><font color="Green" title="E46">A48</font></span></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</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="45_4_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_4_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E47">A49</font></span>: 
<font color="Maroon" title="c13">q2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r2</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="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )</span><br/><div class="add"><a NAME="E2:45_4_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E48">A50</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</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/><span class="kw">consider </span><font color="Maroon" title="c14">q3</font> being    <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="c4">n</font><span class="kw"> such that </span><br/><a NAME="E4:45_4_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E49">A51</font></span>: 
<font color="Maroon" title="c14">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_4_1_1_3"><span class="lab"><font color="Green" title="E36">A38</font></span></a></span>;<br/><a NAME="E5:45_4_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E50">A52</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</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="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E1:45_4_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E47">A49</font></span></a>, <a class="txt" href="#E4:45_4_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E49">A51</font></span></a></span>;<br/><a NAME="E6:45_4_1_1_3_1_1_2"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E11:45_4_1_1_3"><span class="lab"><font color="Green" title="E43">A45</font></span></a>, <a class="txt" href="#E8:45_4_1_1_3"><span class="lab"><font color="Green" title="E40">A42</font></span></a>, <a class="txt" href="#E12:45_4_1_1_3"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="txt" href="#E10:45_4_1_1_3"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a></span>;<br/><a NAME="E7:45_4_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A53</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_4_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E48">A50</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E8:45_4_1_1_3_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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c14">q3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E1:45_4_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E47">A49</font></span></a>, <a class="txt" href="#E4:45_4_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E49">A51</font></span></a></span>;<br/><a NAME="E9:45_4_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A54</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E4:45_4_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E49">A51</font></span></a></span>;<br/><a NAME="E10:45_4_1_1_3_1_1_2"/>
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c14">q3</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="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E1:45_4_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E47">A49</font></span></a>, <a class="txt" href="#E4:45_4_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E49">A51</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:45_4_1_1_3_1_1_2"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E11:45_4_1_1_3"><span class="lab"><font color="Green" title="E43">A45</font></span></a>, <a class="txt" href="#E8:45_4_1_1_3"><span class="lab"><font color="Green" title="E40">A42</font></span></a>, <a class="txt" href="#E12:45_4_1_1_3"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="txt" href="#E10:45_4_1_1_3"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="txt" href="#E4:45_4_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E49">A51</font></span></a>, <a class="txt" href="#E5:45_4_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A52</font></span></a>, <a class="txt" href="#E7:45_4_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E51">A53</font></span></a>, <a class="txt" href="#E9:45_4_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E16:45_4_1_1_3"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_4_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:45_4_1_1_4"/><span class="lab"><font color="Green" title="E36">A55</font></span>: 
( not <font color="Maroon" title="c9">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="c4">n</font> &amp; not <font color="Maroon" title="c10">r2</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="c4">n</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="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )</span><br/><div class="add"><a NAME="E2:45_4_1_1_4"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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="c4">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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="c4">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="urysohn1.html#T20" target="_self" title="URYSOHN1:th.20">Th20</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">q11</font>, <font color="Maroon" title="c13">q21</font> being    <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="c4">n</font><span class="kw"> such that </span><br/><a NAME="E4:45_4_1_1_4"/><span class="lab"><font color="Green" title="E37">A56</font></span>: 
<font color="Maroon" title="c12">q11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:45_4_1_1_4"/><span class="lab"><font color="Green" title="E38">A57</font></span>: 
<font color="Maroon" title="c13">q21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E6:45_4_1_1_4"/>
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_4_1_1_4"><span class="lab"><font color="Green" title="E36">A55</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E7:45_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A58</font></span>: 
<font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r1</font> is    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q11</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E4:45_4_1_1_4"><span class="lab"><font color="Green" title="E37">A56</font></span></a>, <a class="txt" href="#E5:45_4_1_1_4"><span class="lab"><font color="Green" title="E38">A57</font></span></a></span>;<br/><a NAME="E8:45_4_1_1_4"/><span class="lab"><font color="Green" title="E40">A59</font></span>: 
<font color="Maroon" title="c12">q11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">q21</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_4_1_1_4"><span class="lab"><font color="Green" title="E37">A56</font></span></a>, <a class="txt" href="#E5:45_4_1_1_4"><span class="lab"><font color="Green" title="E38">A57</font></span></a>, <a class="ref" href="urysohn1.html#T18" target="_self" title="URYSOHN1:th.18">Th18</a></span>;<br/><a NAME="E9:45_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A60</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</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="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/><a NAME="E10:45_4_1_1_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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="c4">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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="c4">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_4_1_1_4"><span class="lab"><font color="Green" title="E36">A55</font></span></a>, <a class="ref" href="urysohn1.html#T20" target="_self" title="URYSOHN1:th.20">Th20</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">q12</font>, <font color="Maroon" title="c15">q22</font> being    <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="c4">n</font><span class="kw"> such that </span><br/><a NAME="E12:45_4_1_1_4"/><span class="lab"><font color="Green" title="E42">A61</font></span>: 
<font color="Maroon" title="c14">q12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E13:45_4_1_1_4"/><span class="lab"><font color="Green" title="E43">A62</font></span>: 
<font color="Maroon" title="c15">q22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E14:45_4_1_1_4"/>
<font color="Maroon" title="c10">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_4_1_1_4"><span class="lab"><font color="Green" title="E36">A55</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E15:45_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A63</font></span>: 
<font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r2</font> is    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c14">q12</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q22</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E12:45_4_1_1_4"><span class="lab"><font color="Green" title="E42">A61</font></span></a>, <a class="txt" href="#E13:45_4_1_1_4"><span class="lab"><font color="Green" title="E43">A62</font></span></a></span>;<br/><a NAME="E16:45_4_1_1_4"/><span class="lab"><font color="Green" title="E45">A64</font></span>: 
<font color="Maroon" title="c14">q12</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">q22</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:45_4_1_1_4"><span class="lab"><font color="Green" title="E42">A61</font></span></a>, <a class="txt" href="#E13:45_4_1_1_4"><span class="lab"><font color="Green" title="E43">A62</font></span></a>, <a class="ref" href="urysohn1.html#T18" target="_self" title="URYSOHN1:th.18">Th18</a></span>;<br/><a NAME="E17:45_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A65</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c14">q12</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="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/><a NAME="E18:45_4_1_1_4"/><span class="lab"><font color="Green" title="E47">A66</font></span>: 
( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q22</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c14">q12</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q22</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E16:45_4_1_1_4"><span class="lab"><font color="Green" title="E45">A64</font></span></a></span>;<br/><a NAME="E19:45_4_1_1_4"/><span class="lab"><font color="Green" title="E48">A67</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c14">q12</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E29">A6</font></span></a></span>;<br/><a NAME="E20:45_4_1_1_4"/><span class="lab"><font color="Green" title="E49">A68</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q11</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E29">A6</font></span></a></span>;<br/><a NAME="E21:45_4_1_1_4"/><span class="lab"><font color="Green" title="E50">A69</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E1:45_4_1_1_4"><span class="lab"><font color="Green" title="E36">A55</font></span></a></span>;<br/><a NAME="E22:45_4_1_1_4"/><span class="lab"><font color="Green" title="E51">A70</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E1:45_4_1_1_4"><span class="lab"><font color="Green" title="E36">A55</font></span></a></span>;<br/><a NAME="E23:45_4_1_1_4"/><span class="lab"><font color="Green" title="E52">A71</font></span>: 
( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q11</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q11</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E8:45_4_1_1_4"><span class="lab"><font color="Green" title="E40">A59</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E24:45_4_1_1_4"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</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="txt" href="#E1:45"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E22:45_4_1_1_4"><span class="lab"><font color="Green" title="E51">A70</font></span></a>, <a class="txt" href="#E7:45_4_1_1_4"><span class="lab"><font color="Green" title="E39">A58</font></span></a>, <a class="txt" href="#E20:45_4_1_1_4"><span class="lab"><font color="Green" title="E49">A68</font></span></a>, <a class="txt" href="#E9:45_4_1_1_4"><span class="lab"><font color="Green" title="E41">A60</font></span></a>, <a class="txt" href="#E21:45_4_1_1_4"><span class="lab"><font color="Green" title="E50">A69</font></span></a>, <a class="txt" href="#E15:45_4_1_1_4"><span class="lab"><font color="Green" title="E44">A63</font></span></a>, <a class="txt" href="#E19:45_4_1_1_4"><span class="lab"><font color="Green" title="E48">A67</font></span></a>, <a class="txt" href="#E17:45_4_1_1_4"><span class="lab"><font color="Green" title="E46">A65</font></span></a>, <a class="txt" href="#E18:45_4_1_1_4"><span class="lab"><font color="Green" title="E47">A66</font></span></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font></span><br/><a NAME="E25:45_4_1_1_4"/><span class="lab"><font color="Green" title="E53">A72</font></span>: 
<font color="Maroon" title="c13">q21</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">q12</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:45"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="txt" href="#E1:45_4_1_1_4"><span class="lab"><font color="Green" title="E36">A55</font></span></a>, <a class="txt" href="#E5:45_4_1_1_4"><span class="lab"><font color="Green" title="E38">A57</font></span></a>, <a class="txt" href="#E12:45_4_1_1_4"><span class="lab"><font color="Green" title="E42">A61</font></span></a>, <a class="ref" href="urysohn1.html#T23" target="_self" title="URYSOHN1:th.23">Th23</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_4_1_1_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_4_1_1_4_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="E1:45_4_1_1_4_1_1"/>
( <font color="Maroon" title="c13">q21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q12</font> or <font color="Maroon" title="c13">q21</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">q12</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:45_4_1_1_4"><span class="lab"><font color="Green" title="E53">A72</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_4_1_1_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_4_1_1_4_1_1_1"/><span class="lab"><font color="Green" title="E54">A73</font></span>: 
<font color="Maroon" title="c13">q21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q12</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font></span><br/><div class="add"><a NAME="E2:45_4_1_1_4_1_1_1"/>
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E22:45_4_1_1_4"><span class="lab"><font color="Green" title="E51">A70</font></span></a>, <a class="txt" href="#E7:45_4_1_1_4"><span class="lab"><font color="Green" title="E39">A58</font></span></a>, <a class="txt" href="#E20:45_4_1_1_4"><span class="lab"><font color="Green" title="E49">A68</font></span></a>, <a class="txt" href="#E9:45_4_1_1_4"><span class="lab"><font color="Green" title="E41">A60</font></span></a>, <a class="txt" href="#E23:45_4_1_1_4"><span class="lab"><font color="Green" title="E52">A71</font></span></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></span>;<br/><a NAME="E3:45_4_1_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A74</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E4:45_4_1_1_4_1_1_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E9:45_4_1_1_4"><span class="lab"><font color="Green" title="E41">A60</font></span></a>, <a class="txt" href="#E21:45_4_1_1_4"><span class="lab"><font color="Green" title="E50">A69</font></span></a>, <a class="txt" href="#E15:45_4_1_1_4"><span class="lab"><font color="Green" title="E44">A63</font></span></a>, <a class="txt" href="#E19:45_4_1_1_4"><span class="lab"><font color="Green" title="E48">A67</font></span></a>, <a class="txt" href="#E18:45_4_1_1_4"><span class="lab"><font color="Green" title="E47">A66</font></span></a>, <a class="txt" href="#E1:45_4_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E54">A73</font></span></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E5:45_4_1_1_4_1_1_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_4_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E55">A74</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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="45_4_1_1_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_4_1_1_4_1_1_2"/><span class="lab"><font color="Green" title="E54">A75</font></span>: 
<font color="Maroon" title="c13">q21</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">q12</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font></span><br/><div class="add"><a NAME="E2:45_4_1_1_4_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="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E22:45_4_1_1_4"><span class="lab"><font color="Green" title="E51">A70</font></span></a>, <a class="txt" href="#E7:45_4_1_1_4"><span class="lab"><font color="Green" title="E39">A58</font></span></a>, <a class="txt" href="#E20:45_4_1_1_4"><span class="lab"><font color="Green" title="E49">A68</font></span></a>, <a class="txt" href="#E9:45_4_1_1_4"><span class="lab"><font color="Green" title="E41">A60</font></span></a>, <a class="txt" href="#E23:45_4_1_1_4"><span class="lab"><font color="Green" title="E52">A71</font></span></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></span>;<br/><a NAME="E3:45_4_1_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A76</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E4:45_4_1_1_4_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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c14">q12</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E1:45_4_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E54">A75</font></span></a></span>;<br/><a NAME="E5:45_4_1_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A77</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c14">q12</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_4_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E6:45_4_1_1_4_1_1_2"/>
( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c14">q12</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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c14">q12</font></span>)</span> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c14">q12</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E21:45_4_1_1_4"><span class="lab"><font color="Green" title="E50">A69</font></span></a>, <a class="txt" href="#E15:45_4_1_1_4"><span class="lab"><font color="Green" title="E44">A63</font></span></a>, <a class="txt" href="#E19:45_4_1_1_4"><span class="lab"><font color="Green" title="E48">A67</font></span></a>, <a class="txt" href="#E17:45_4_1_1_4"><span class="lab"><font color="Green" title="E46">A65</font></span></a>, <a class="txt" href="#E18:45_4_1_1_4"><span class="lab"><font color="Green" title="E47">A66</font></span></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></span>;<br/><a NAME="E7:45_4_1_1_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c14">q12</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font>
 <span class="kw">by</span> <span class="lab"><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="E8:45_4_1_1_4_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="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_4_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E56">A77</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E27:45_4_1_1_4"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:45_4"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</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">thus </span><a NAME="E19:45"/>
( <font color="Maroon" title="c11">r</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="c4">n</font> implies <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45"><span class="lab"><font color="Green" title="E34">A17</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>
