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

<b>let </b><font color="Maroon" title="c1">D</font> be  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> ;<br/><b>let </b><font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c3">H2</font> be    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">D</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/>







<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E30">A1</font></i>: 
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c2">H1</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c3">H2</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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> ;<br/><a NAME="E1:45_1"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E2:45_1"/>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:45"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="seqfunc.html#D10" target="_self" title="SEQFUNC:def.10">Def10</a></i>;<br/><a NAME="E3:45_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:45_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:45_1"/><b>then </b><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D11" target="_self" title="SEQFUNC:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D11" target="_self" title="SEQFUNC:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="seqfunc.html#E4:45_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="seqfunc.html#D6" target="_self" title="SEQFUNC:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="seqfunc.html#D11" target="_self" title="SEQFUNC:def.11">Def11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:45"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K6" title="SEQFUNC:func.6">+</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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> ;<br/><a NAME="E1:45_2"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E2:45_2"/>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:45"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="seqfunc.html#D10" target="_self" title="SEQFUNC:def.10">Def10</a></i>;<br/><a NAME="E3:45_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:45_2"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:45_2"/><b>then </b><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6" title="RFUNCT_2:th.6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D11" target="_self" title="SEQFUNC:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D11" target="_self" title="SEQFUNC:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="seqfunc.html#E4:45_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="seqfunc.html#T4" target="_self" title="SEQFUNC:th.4">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="seqfunc.html#D11" target="_self" title="SEQFUNC:def.11">Def11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:45"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K7" title="SEQFUNC:func.7">-</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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> ;<br/><a NAME="E1:45_3"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E2:45_3"/>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:45"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="seqfunc.html#D10" target="_self" title="SEQFUNC:def.10">Def10</a></i>;<br/><a NAME="E3:45_3"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="seqfunc.html#E1:45_3"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:45_3"/><b>then </b><i><font color="Green" title="E32">A7</font></i>: 
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#D5" title="SEQ_1:def.5">SEQ_1:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D11" target="_self" title="SEQFUNC:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D11" target="_self" title="SEQFUNC:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="seqfunc.html#E4:45_3"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="seq_1.html#D5" title="SEQ_1:def.5">SEQ_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="seqfunc.html#D8" target="_self" title="SEQFUNC:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="seqfunc.html#D11" target="_self" title="SEQFUNC:def.11">Def11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:45"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H2</font> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K8" title="SEQFUNC:func.8">(#)</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="seqfunc.html#K10" title="SEQFUNC:func.10">#</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
