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

<b>set </b><font color="Maroon" title="c5">XX</font> = the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b1">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b2">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">L</font> &amp;  ex <font color="Olive" title="b3">X</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">X</font> &amp; <font color="Olive" title="b1">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b3">X</font> ) );<br/>
<a NAME="E1:20_1_1"/><i><font color="Green" title="E14">A2</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>
 
;<br/>
<a NAME="E2:20_1_1"/><i><font color="Green" title="E15">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</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> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y1</font>, <font color="Maroon" title="c8">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:20_1_1_1"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>
 <b>and </b><br/><a NAME="E2:20_1_1_1"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y1</font>]
 <b>and </b><br/><a NAME="E3:20_1_1_1"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">y2</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">y1</font> = <font color="Maroon" title="c7">y1</font>, <font color="Maroon" title="c10">y2</font> = <font color="Maroon" title="c8">y2</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E2:20_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="projred1.html#E3:20_1_1_1"><i><font color="Green" title="E18">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c6">x</font> as   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> <b>by </b><i><a class="txt" href="projred1.html#E1:20_1_1_1"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">A1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="incsp_1.html#U2" title="INCSP_1:sel.2">Lines</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E7:20_1_1_1"/><i><font color="Green" title="E19">A7</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">A1</font> &amp; <font color="Maroon" title="c11">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">A1</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">A1</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E2:20_1_1_1"><i><font color="Green" title="E17">A5</font></i></a></i>;<br/>
<a NAME="E8:20_1_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">A1</font> &amp; <font color="Maroon" title="c11">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">A1</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c12">A1</font> )
 
<b>by </b><i><a class="txt" href="projred1.html#E1:20"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="projred1.html#E3:20_1_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="projred1.html#E7:20_1_1_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>
<b>hence </b><a NAME="E9:20_1_1_1"/>
<font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y2</font>
 <b>by </b><i><a class="txt" href="projred1.html#E1:20"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="projred1.html#E2:20_1_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="projred1.html#E3:20_1_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="projred1.html#E7:20_1_1_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="incproj.html#D9" title="INCPROJ:def.9">INCPROJ:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>,the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E4:20_1_1"/><i><font color="Green" title="E16">A8</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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font> &amp;  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>and </b><br/><a NAME="E5:20_1_1"/><i><font color="Green" title="E17">A9</font></i>: 
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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="partfun1.html#S2" title="PARTFUN1:sch.2">PARTFUN1:sch 2</a>(<a class="txt" href="projred1.html#E1:20_1_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="projred1.html#E2:20_1_1"><i><font color="Green" title="E15">A3</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c6">f</font>
;<br/>

<b>thus </b><a NAME="E6:20_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="incsp_1.html#U1" title="INCSP_1:sel.1">Points</a> of <font color="Maroon" title="c1">IPP</font>
 ;<br/>

<b>thus </b><a NAME="E7:20_1_1"/><i><font color="Green" title="E18">A10</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> iff <font color="Olive" title="b1">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>;<br/>

<a NAME="E1:20_1_1_2"/><i><font color="Green" title="E19">A11</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> implies <font color="Maroon" title="c7">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 ;<br/>

<a NAME="E2:20_1_1_2_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>]
 
<b>by </b><i><a class="txt" href="projred1.html#E5:20_1_1"><i><font color="Green" title="E17">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:20_1_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font> implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_2_2"/><i><font color="Green" title="E20">A12</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">X</font> being   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font><b> such that </b><br/><a NAME="E3:20_1_1_2_2"/><i><font color="Green" title="E21">A13</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">X</font> &amp; <font color="Maroon" title="c7">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">X</font> )
 <b>by </b><i><a class="ref" href="incproj.html#D10" title="INCPROJ:def.10">INCPROJ:def 10</a></i>;<br/>
<a NAME="E4:20_1_1_2_2"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Olive" title="b1">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">X</font> )
 
<b>by </b><i><a class="ref" href="incproj.html#D14" title="INCPROJ:def.14">INCPROJ:def 14</a></i>;<br/>
<b>hence </b><a NAME="E5:20_1_1_2_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="projred1.html#E4:20_1_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="projred1.html#E1:20_1_1_2_2"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="projred1.html#E3:20_1_1_2_2"><i><font color="Green" title="E21">A13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> iff <font color="Maroon" title="c7">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:20_1_1_2"><i><font color="Green" title="E19">A11</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <font color="Maroon" title="c1">IPP</font>;<br/>



<b>assume </b><a NAME="E8:20_1_1"/><i><font color="Green" title="E19">A14</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c8">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c3">L</font> )
 ;<br/>

<b>thus </b><a NAME="E9:20_1_1"/>
( <font color="Maroon" title="c6">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="c8">y</font> iff  ex <font color="Olive" title="b1">X</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c7">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c8">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_3"/><i><font color="Green" title="E20">A15</font></i>: 
( <font color="Maroon" title="c6">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="c8">y</font> implies  ex <font color="Olive" title="b1">X</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c7">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c8">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_3_1"/><i><font color="Green" title="E21">A16</font></i>: 
<font color="Maroon" title="c6">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="c8">y</font>
 ;<br/>

<a NAME="E2:20_1_1_3_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E7:20_1_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="projred1.html#E8:20_1_1"><i><font color="Green" title="E19">A14</font></i></a></i>;<br/>
<a NAME="E3:20_1_1_3_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>]
 
<b>by </b><i><a class="txt" href="projred1.html#E5:20_1_1"><i><font color="Green" title="E17">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:20_1_1_3_1"/>
 ex <font color="Olive" title="b1">X</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c7">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c8">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> )
 <b>by </b><i><a class="txt" href="projred1.html#E1:20_1_1_3_1"><i><font color="Green" title="E21">A16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_1_1_3"/>
(  ex <font color="Olive" title="b1">X</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c7">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c8">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> ) implies <font color="Maroon" title="c6">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="c8">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_3_2"/><i><font color="Green" title="E21">A17</font></i>: 
 ex <font color="Olive" title="b1">X</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c7">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c8">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> )
 ;<br/>

<a NAME="E2:20_1_1_3_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="projred1.html#E7:20_1_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="projred1.html#E8:20_1_1"><i><font color="Green" title="E19">A14</font></i></a></i>;<br/>
<a NAME="E3:20_1_1_3_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>]
 
<b>by </b><i><a class="txt" href="projred1.html#E5:20_1_1"><i><font color="Green" title="E17">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:20_1_1_3_2"/>
<font color="Maroon" title="c6">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="c8">y</font>
 <b>by </b><i><a class="txt" href="projred1.html#E2:20_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="projred1.html#E8:20_1_1"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="projred1.html#E1:20_1_1_3_2"><i><font color="Green" title="E21">A17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1_1_3"/>
( <font color="Maroon" title="c6">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="c8">y</font> iff  ex <font color="Olive" title="b1">X</font> being  <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <font color="Maroon" title="c1">IPP</font> st <br/>( <font color="Maroon" title="c4">p</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c7">x</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c8">y</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Olive" title="b1">X</font> ) )
 <b>by </b><i><a class="txt" href="projred1.html#E1:20_1_1_3"><i><font color="Green" title="E20">A15</font></i></a></i>;<br/>


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


</div>
