<?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">M</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">R</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : <font color="Olive" title="b1">f</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="c2">M</font> </span>}</span> ;<br/>
<a NAME="E1:6"/><i><font color="Green" title="E3">A1</font></i>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:6_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c2">M</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_1_1"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>and </b><br/><a NAME="E4:6_1_1"/><i><font color="Green" title="E5">A3</font></i>: 
<font color="Maroon" title="c5">Y</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> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c2">M</font></span>)</span>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:6"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:6_1_1"/><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon" title="c6">i</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> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c2">M</font></span>)</span>
 <b>and </b><br/><a NAME="E7:6_1_1"/><i><font color="Green" title="E7">A5</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E4:6_1_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:6_1_1"/><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E6:6_1_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="funct_6.html#T90" title="FUNCT_6:th.90">FUNCT_6:90</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c6">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font> <b>by </b><i><a class="ref" href="gobrd13.html#T1" target="_self" title="GOBRD13:th.1">Th1</a></i>;<br/><a NAME="E11:6_1_1"/>
( <font color="Maroon" title="c8">f</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="c2">M</font> &amp; <font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E8:6_1_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:6_1_1"/><b>then </b><i><font color="Green" title="E9">A7</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : <font color="Olive" title="b1">f</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="c2">M</font> </span>}</span> 
 ;<br/><a NAME="E13:6_1_1"/>
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:6_1_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="gobrd13.html#E8:6_1_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="funct_6.html#T31" title="FUNCT_6:th.31">FUNCT_6:31</a></i>;<br/><b>hence </b><a NAME="E14:6_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : <font color="Olive" title="b1">f</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="c2">M</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:6_1_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="gobrd13.html#E12:6_1_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:6_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : <font color="Olive" title="b1">f</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="c2">M</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c5">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:6_1"/><i><font color="Green" title="E4">A8</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>and </b><br/><a NAME="E5:6_1"/><i><font color="Green" title="E5">A9</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : <font color="Olive" title="b1">f</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="c2">M</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">f</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E7:6_1"/><i><font color="Green" title="E6">A10</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font>
 <b>and </b><br/><a NAME="E8:6_1"/><i><font color="Green" title="E7">A11</font></i>: 
<font color="Maroon" title="c6">f</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="c2">M</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E5:6_1"><i><font color="Green" title="E5">A9</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:6_1"/><i><font color="Green" title="E8">A12</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">M</font>
 <b>and </b><br/><a NAME="E11:6_1"/><i><font color="Green" title="E9">A13</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E8:6_1"><i><font color="Green" title="E7">A11</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><a NAME="E12:6_1"/>
( <font color="Maroon" title="c7">i</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> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c2">M</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E10:6_1"><i><font color="Green" title="E8">A12</font></i></a>, <a class="txt" href="gobrd13.html#E11:6_1"><i><font color="Green" title="E9">A13</font></i></a>, <a class="ref" href="funct_6.html#T31" title="FUNCT_6:th.31">FUNCT_6:31</a></i>;<br/><a NAME="E13:6_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</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> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c2">M</font></span>)</span>
 <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>hence </b><a NAME="E14:6_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:6"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="gobrd13.html#E4:6_1"><i><font color="Green" title="E4">A8</font></i></a>, <a class="txt" href="gobrd13.html#E7:6_1"><i><font color="Green" title="E6">A10</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:6"/>
 <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : <font color="Olive" title="b1">f</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="c2">M</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div>
