<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be  <a href="membered.html#V3" title="MEMBERED:attr.3">real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <font color="Maroon" title="c2">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:1"/><i><font color="Green" title="E2">A1</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>from </b><i><a class="ref" href="graph_5.html#S1" title="GRAPH_5:sch.1">GRAPH_5:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c3">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a><br/>

<a NAME="E3:1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:1_1"/><i><font color="Green" title="E4">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )
 ;<br/>

<a NAME="E2:1_1"/>
( <font color="Maroon" title="c4">x</font> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> &amp; <font color="Maroon" title="c5">y</font> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> )
 
<b>by </b><i><a class="txt" href="int_4.html#E2:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="int_4.html#E1:1_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="membered.html#D3" title="MEMBERED:def.3">MEMBERED:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E4:1_1"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">y</font> )
 
<b>by </b><i><a class="txt" href="int_4.html#E2:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="int_4.html#E1:1_1"><i><font color="Green" title="E4">A3</font></i></a></i>;<br/>
<a NAME="E5:1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">y</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:1_1"><i><font color="Green" title="E4">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">X</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_2_1"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:1_2_1"/><i><font color="Green" title="E4">A4</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="int_4.html#E2:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="int_4.html#E3:1_2_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="membered.html#D3" title="MEMBERED:def.3">MEMBERED:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E6:1_2_1"/>
<font color="Maroon" title="c2">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">x</font> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>
 
;<br/>
<a NAME="E7:1_2_1"/><b>then </b>
<font color="Maroon" title="c4">z</font> is <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>
 
<b>by </b><i><a class="txt" href="int_4.html#E2:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="int_4.html#E3:1_2_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">z'</font> = <font color="Maroon" title="c4">z</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E9:1_2_1"/>
<font color="Maroon" title="c8">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">x</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E2:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="int_4.html#E3:1_2_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:1_2_1"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="int_4.html#E2:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="int_4.html#E3:1_2_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="measure6.html#D6" title="MEASURE6:def.6">MEASURE6:def 6</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:1_2"/><i><font color="Green" title="E4">A5</font></i>: 
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c4">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c6">x</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:1_2"/><i><font color="Green" title="E5">A6</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="int_4.html#E2:1_2"><i><font color="Green" title="E4">A5</font></i></a>, <a class="ref" href="measure6.html#D6" title="MEASURE6:def.6">MEASURE6:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E7:1_2"/><i><font color="Green" title="E6">A7</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> )
 
<b>by </b><i><a class="txt" href="int_4.html#E2:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="int_4.html#E5:1_2"><i><font color="Green" title="E5">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E9:1_2"/><i><font color="Green" title="E7">A8</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="int_4.html#E7:1_2"><i><font color="Green" title="E6">A7</font></i></a></i>;<br/>
<b>thus </b><a NAME="E10:1_2"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="int_4.html#E9:1_2"><i><font color="Green" title="E7">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:1"/>
( <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="txt" href="int_4.html#E2:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="int_4.html#E3:1"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>


</div>
