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<b>let </b><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#V10" title="PRE_TOPC:attr.10">normal</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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>  &amp; <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> holds <br/> ex <font color="Olive" title="b3">F</font> being  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_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/>for <font color="Olive" title="b4">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b3">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b4">n</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>,<font color="Olive" title="b4">n</font> &amp; ( for <font color="Olive" title="b5">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">r</font> ) )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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>  &amp; <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> implies  ex <font color="Olive" title="b1">F</font> being  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_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/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b1">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b2">n</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b2">n</font> &amp; ( for <font color="Olive" title="b3">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">r</font> ) ) )</span><br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E7">A1</font></i>: 
( <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>  &amp; <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">F</font> being  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_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/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b1">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b2">n</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b2">n</font> &amp; ( for <font color="Olive" title="b3">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">r</font> ) )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st $1 is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">n</font>;<br/>
<b>consider </b><font color="Maroon" title="c4">D</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:7"/><i><font color="Green" title="E8">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_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> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c5">S</font> being    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<a NAME="E5:7"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c5">S</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_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>
 
<b>by </b><i><a class="ref" href="urysohn3.html#T4" target="_self" title="URYSOHN3:th.4">Th4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">D</font> = <font color="Maroon" title="c4">D</font> 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>  <b>by </b><i><a class="txt" href="#E3:7"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">S</font> = <font color="Maroon" title="c5">S</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="txt" href="#E3:7"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="#E5:7"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">D</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">D</font>] <b>means </b> ex <font color="Olive" title="b1">xx</font>, <font color="Olive" title="b2">yy</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> , <a href="pcomps_1.html#K1" title="PCOMPS_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> st <br/>( <font color="Olive" title="b1">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">yy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp;  ex <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Olive" title="b1">xx</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b3">k</font> &amp; ( for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">xx</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b3">k</font> holds <br/><font color="Olive" title="b2">yy</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) &amp; ( for <font color="Olive" title="b3">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">xx</font> holds  <font color="Olive" title="b1">xx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">yy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">r</font> ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">D</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">D</font>] <b>means </b><font color="Maroon">S<sub>2</sub></font>[$2,$3];<br/>
<a NAME="E8:7"/><i><font color="Green" title="E10">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">D</font>  ex <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">D</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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">D</font>  ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">D</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span><br/><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">D</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>



<b>consider </b><font color="Maroon" title="c10">s0</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:7_1"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c9">x</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c10">s0</font>
 <b>by </b><i><a class="txt" href="#E3:7"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">xx</font> = <font color="Maroon" title="c9">x</font> as    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c10">s0</font> <b>by </b><i><a class="txt" href="#E2:7_1"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">yy</font> being    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c10">s0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1<b> such that </b><br/><a NAME="E5:7_1"/><i><font color="Green" title="E12">A6</font></i>: 
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="c10">s0</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>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c10">s0</font> holds <br/><font color="Maroon" title="c12">yy</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font>
 <b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="urysohn3.html#T3" target="_self" title="URYSOHN3:th.3">Th3</a></i>;<br/>
<a NAME="E6:7_1"/><i><font color="Green" title="E13">A7</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c11">xx</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">k</font> holds <br/><font color="Maroon" title="c12">yy</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">xx</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c13">k</font> implies <font color="Maroon" title="c12">yy</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/>

<b>assume </b><a NAME="E1:7_1_1"/>
<font color="Maroon" title="c11">xx</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c13">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">yy</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/>

<a NAME="E2:7_1_1"/><b>then </b><i><font color="Green" title="E14">A8</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c13">k</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:7_1_1"/>
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1_1"/><i><font color="Green" title="E15">A9</font></i>: 
<font color="Maroon" title="c13">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">s0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_1_1_1"/>
( <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">s0</font> or <font color="Maroon" title="c10">s0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">k</font> )
 </span><b>by </b><i><a class="txt" href="#E1:7_1_1_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_1"/><i><font color="Green" title="E16">A10</font></i>: 
<font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">s0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>set </b><font color="Maroon" title="c14">o</font> = 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font></span>)</span>;<br/><a NAME="E2:7_1_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font> &amp; 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font></span>)</span> )
 <b>by </b><i><a class="ref" href="prepower.html#T19" title="PREPOWER:th.19">PREPOWER:19</a></i>;<br/><a NAME="E3:7_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E17">A11</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font></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="c10">s0</font>
 <b>by </b><i><a class="ref" href="urysohn1.html#D3" title="URYSOHN1:def.3">URYSOHN1:def 3</a></i>;<br/><a NAME="E4:7_1_1_1_1_1"/>
not 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font></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="c13">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1_1_1_1_1"/>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font></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="c13">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c15">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:7_1_1_1_1_1_1"/><i><font color="Green" title="E18">A12</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font> &amp; 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="urysohn1.html#D3" title="URYSOHN1:def.3">URYSOHN1:def 3</a></i>;<br/>
<a NAME="E4:7_1_1_1_1_1_1"/><i><font color="Green" title="E19">A13</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font> )
 
<b>by </b><i><a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a></i>;<br/>
<a NAME="E5:7_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E20">A14</font></i>: 
1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:7_1_1_1_1_1_1"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="xcmplx_1.html#T96" title="XCMPLX_1:th.96">XCMPLX_1:96</a></i>;<br/>
<a NAME="E6:7_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E21">A15</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:7_1_1_1_1_1_1"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E7:7_1_1_1_1_1_1"/><i><font color="Green" title="E22">A16</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E5:7_1_1_1_1_1_1"><i><font color="Green" title="E20">A14</font></i></a>, <a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a></i>;<br/>
<a NAME="E8:7_1_1_1_1_1_1"/>
2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:7_1_1_1_1_1"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="pepin.html#T71" title="PEPIN:th.71">PEPIN:71</a></i>;<br/>
<a NAME="E9:7_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E23">A17</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="#E6:7_1_1_1_1_1_1"><i><font color="Green" title="E21">A15</font></i></a>, <a class="ref" href="xreal_1.html#T85" title="XREAL_1:th.85">XREAL_1:85</a></i>;<br/>
<a NAME="E10:7_1_1_1_1_1_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds not <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c16">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E2:7_1_1_1_1_1_1_1"/><i><font color="Green" title="E24">A18</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:7_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E9:7_1_1_1_1_1_1"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="#E2:7_1_1_1_1_1_1_1"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="xreal_1.html#T51" title="XREAL_1:th.51">XREAL_1:51</a></i>;<br/>
<b>hence </b><a NAME="E4:7_1_1_1_1_1_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:7_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:7_1_1_1_1_1_1"><i><font color="Green" title="E22">A16</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:7_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:7_1_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="#E3:7_1_1_1_1_1"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_2"/><i><font color="Green" title="E16">A19</font></i>: 
<font color="Maroon" title="c10">s0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>set </b><font color="Maroon" title="c14">o</font> = 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span>;<br/><a NAME="E2:7_1_1_1_1_2"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font> &amp; 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="prepower.html#T19" title="PREPOWER:th.19">PREPOWER:19</a></i>;<br/><a NAME="E3:7_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E17">A20</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></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="c13">k</font>
 <b>by </b><i><a class="ref" href="urysohn1.html#D3" title="URYSOHN1:def.3">URYSOHN1:def 3</a></i>;<br/><a NAME="E4:7_1_1_1_1_2"/>
not 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></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="c10">s0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1_1_1_2_1"/>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></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="c10">s0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c15">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:7_1_1_1_1_2_1"/><i><font color="Green" title="E18">A21</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font> &amp; 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font></span>)</span> )
 <b>by </b><i><a class="ref" href="urysohn1.html#D3" title="URYSOHN1:def.3">URYSOHN1:def 3</a></i>;<br/>
<a NAME="E4:7_1_1_1_1_2_1"/><i><font color="Green" title="E19">A22</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font> )
 
<b>by </b><i><a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a></i>;<br/>
<a NAME="E5:7_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E20">A23</font></i>: 
1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:7_1_1_1_1_2_1"><i><font color="Green" title="E18">A21</font></i></a>, <a class="ref" href="xcmplx_1.html#T96" title="XCMPLX_1:th.96">XCMPLX_1:96</a></i>;<br/>
<a NAME="E6:7_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E21">A24</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font></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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:7_1_1_1_1_2_1"><i><font color="Green" title="E19">A22</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E7:7_1_1_1_1_2_1"/><i><font color="Green" title="E22">A25</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E5:7_1_1_1_1_2_1"><i><font color="Green" title="E20">A23</font></i></a>, <a class="ref" href="newton.html#T102" title="NEWTON:th.102">NEWTON:102</a></i>;<br/>
<a NAME="E8:7_1_1_1_1_2_1"/>
2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c10">s0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:7_1_1_1_1_2"><i><font color="Green" title="E16">A19</font></i></a>, <a class="ref" href="pepin.html#T71" title="PEPIN:th.71">PEPIN:71</a></i>;<br/>
<a NAME="E9:7_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E23">A26</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="#E6:7_1_1_1_1_2_1"><i><font color="Green" title="E21">A24</font></i></a>, <a class="ref" href="xreal_1.html#T85" title="XREAL_1:th.85">XREAL_1:85</a></i>;<br/>
<a NAME="E10:7_1_1_1_1_2_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds not <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c16">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E2:7_1_1_1_1_2_1_1"/><i><font color="Green" title="E24">A27</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:7_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E9:7_1_1_1_1_2_1"><i><font color="Green" title="E23">A26</font></i></a>, <a class="txt" href="#E2:7_1_1_1_1_2_1_1"><i><font color="Green" title="E24">A27</font></i></a>, <a class="ref" href="xreal_1.html#T51" title="XREAL_1:th.51">XREAL_1:51</a></i>;<br/>
<b>hence </b><a NAME="E4:7_1_1_1_1_2_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:7_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:7_1_1_1_1_2_1"><i><font color="Green" title="E22">A25</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:7_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:7_1_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="#E3:7_1_1_1_1_2"><i><font color="Green" title="E17">A20</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:7_1_1"/>
<font color="Maroon" title="c12">yy</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:7_1"/><i><font color="Green" title="E14">A28</font></i>: 
for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">xx</font> holds  <font color="Maroon" title="c11">xx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">yy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">xx</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">xx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">yy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">r</font></span><br/>

<a NAME="E1:7_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c10">s0</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:7_1_2"/><b>then </b><i><font color="Green" title="E15">A29</font></i>: 
<font color="Maroon" title="c13">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="c10">s0</font>
 
;<br/>
<a NAME="E3:7_1_2"/>
 <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c10">s0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="urysohn1.html#T11" title="URYSOHN1:th.11">URYSOHN1:11</a></i>;<br/>
<b>hence </b><a NAME="E4:7_1_2"/>
<font color="Maroon" title="c11">xx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">yy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">r</font>
 <b>by </b><i><a class="txt" href="#E5:7_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="#E2:7_1_2"><i><font color="Green" title="E15">A29</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c13">xx</font> = <font color="Maroon" title="c11">xx</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_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> <b>by </b><i><a class="ref" href="urysohn3.html#T4" target="_self" title="URYSOHN3:th.4">Th4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">xx</font> = <font color="Maroon" title="c13">xx</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">D</font> ;<br/>
<a NAME="E10:7_1"/><i><font color="Green" title="E15">A30</font></i>: 
<font color="Maroon" title="c12">yy</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_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>
 
<b>by </b><i><a class="ref" href="urysohn3.html#T4" target="_self" title="URYSOHN3:th.4">Th4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">yy</font> = <font color="Maroon" title="c12">yy</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="txt" href="#E3:7"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<a NAME="E12:7_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">D</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_3"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c15">yy</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c15">yy</font>]</span><br/>

<b>reconsider </b><font color="Maroon" title="c16">xx</font> = <font color="Maroon" title="c14">xx</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> , <a href="pcomps_1.html#K1" title="PCOMPS_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> <b>by </b><i><a class="ref" href="partfun1.html#T121" title="PARTFUN1:th.121">PARTFUN1:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">yy</font> = <font color="Maroon" title="c15">yy</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> , <a href="pcomps_1.html#K1" title="PCOMPS_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> <b>by </b><i><a class="txt" href="#E10:7_1"><i><font color="Green" title="E15">A30</font></i></a>, <a class="ref" href="partfun1.html#T120" title="PARTFUN1:th.120">PARTFUN1:120</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c16">xx</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">yy</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> , <a href="pcomps_1.html#K1" title="PCOMPS_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> st <br/>( <font color="Maroon" title="c16">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Olive" title="b1">yy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">yy</font> &amp;  ex <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">xx</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b2">k</font> &amp; ( for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c16">xx</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b2">k</font> holds <br/><font color="Olive" title="b1">yy</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) &amp; ( for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">xx</font> holds  <font color="Maroon" title="c16">xx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">yy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c17">yy</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c17">yy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">yy</font> &amp;  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">xx</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">k</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c16">xx</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">k</font> holds <br/><font color="Maroon" title="c17">yy</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) &amp; ( for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">xx</font> holds  <font color="Maroon" title="c16">xx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">yy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> ) )</span><br/>

<b>thus </b><a NAME="E3:7_1_3"/>
( <font color="Maroon" title="c16">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c17">yy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">yy</font> &amp;  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c16">xx</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">k</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c16">xx</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">k</font> holds <br/><font color="Maroon" title="c17">yy</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) &amp; ( for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">xx</font> holds  <font color="Maroon" title="c16">xx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">yy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> ) )
 <b>by </b><i><a class="txt" href="#E6:7_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="#E7:7_1"><i><font color="Green" title="E14">A28</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c16">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">D</font><b> such that </b><br/><a NAME="E14:7_1"/><i><font color="Green" title="E16">A31</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">y</font>]
 ;<br/>
<b>take </b>
<font color="Maroon" title="c16">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">y</font>]</span><br/>

<b>thus </b><a NAME="E15:7_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c16">y</font>]
 <b>by </b><i><a class="txt" href="#E14:7_1"><i><font color="Green" title="E16">A31</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:7"/>
 ex <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c6">D</font> st <br/>( <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font>,<font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>] ) )
 
<b>from </b><i><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="#E8:7"><i><font color="Green" title="E10">A4</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c8">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c6">D</font><b> such that </b><br/><a NAME="E11:7"/><i><font color="Green" title="E11">A32</font></i>: 
( <font color="Maroon" title="c8">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c8">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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>] ) )
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> , <a href="pcomps_1.html#K1" title="PCOMPS_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>,  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> , <a href="pcomps_1.html#K1" title="PCOMPS_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>] <b>means </b>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st $2 is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">k</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st $2 is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">k</font> holds <br/>$3 is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) &amp; ( for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> $2 holds  $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> ) );<br/>
<a NAME="E12:7"/><i><font color="Green" title="E12">A33</font></i>: 
( <font color="Maroon" title="c8">F</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">D</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E13:7"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_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>
 
<b>by </b><i><a class="txt" href="#E3:7"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<a NAME="E14:7"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_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>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E15:7"/><b>then </b>
( <font color="Maroon" title="c8">F</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_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> )
 
<b>by </b><i><a class="txt" href="#E12:7"><i><font color="Green" title="E12">A33</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c8">F</font> as   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_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> <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> $1 is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,$1 &amp; ( for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> $1</span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> ) );<br/>
<a NAME="E17:7"/>
 ex <font color="Olive" title="b1">xx</font>, <font color="Olive" title="b2">yy</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> , <a href="pcomps_1.html#K1" title="PCOMPS_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> st <font color="Maroon">S<sub>4</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive" title="b1">xx</font>,<font color="Olive" title="b2">yy</font>]
 
<b>by </b><i><a class="txt" href="#E11:7"><i><font color="Green" title="E11">A32</font></i></a></i>;<br/>
<a NAME="E18:7"/><b>then </b><i><font color="Green" title="E13">A34</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="#E11:7"><i><font color="Green" title="E11">A32</font></i></a></i>;<br/>
<a NAME="E19:7"/><i><font color="Green" title="E14">A35</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c10">n</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:7_2"/><i><font color="Green" title="E15">A36</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c10">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<b>consider </b><font color="Maroon" title="c11">xx</font>, <font color="Maroon" title="c12">yy</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> , <a href="pcomps_1.html#K1" title="PCOMPS_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><b> such that </b><br/><a NAME="E3:7_2"/><i><font color="Green" title="E16">A37</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c10">n</font>,<font color="Maroon" title="c11">xx</font>,<font color="Maroon" title="c12">yy</font>]
 <b>by </b><i><a class="txt" href="#E11:7"><i><font color="Green" title="E11">A32</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:7_2"/>
<font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E1:7_2"><i><font color="Green" title="E15">A36</font></i></a>, <a class="txt" href="#E3:7_2"><i><font color="Green" title="E16">A37</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font></span><br/>

<b>let </b><font color="Maroon" title="c13">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">r</font></span><br/>

<b>consider </b><font color="Maroon" title="c14">xx1</font>, <font color="Maroon" title="c15">yy1</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> , <a href="pcomps_1.html#K1" title="PCOMPS_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><b> such that </b><br/><a NAME="E6:7_2"/><i><font color="Green" title="E17">A38</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c14">xx1</font>,<font color="Maroon" title="c15">yy1</font>]
 <b>by </b><i><a class="txt" href="#E11:7"><i><font color="Green" title="E11">A32</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:7_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">r</font>
 <b>by </b><i><a class="txt" href="#E6:7_2"><i><font color="Green" title="E17">A38</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E20:7"/><i><font color="Green" title="E15">A39</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E18:7"><i><font color="Green" title="E13">A34</font></i></a>, <a class="txt" href="#E19:7"><i><font color="Green" title="E14">A35</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c9">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b1">n</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">n</font> &amp; ( for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> ) )</span><br/>

<b>thus </b><a NAME="E21:7"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b1">n</font> is    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">n</font> &amp; ( for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="urysohn3.html#K1" title="URYSOHN3:func.1">.</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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> ) )
 <b>by </b><i><a class="txt" href="#E20:7"><i><font color="Green" title="E15">A39</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
