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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="ordinal1.html#V6">c=-linear</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="rfunct_3.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>



<a NAME="E1:10"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="trees_2.html#T36">TREES_2:36</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>suppose </b></a><a NAME="E1:10_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="hahnban.html#T4" target="_self">Th4</a></i>;<br/><div><i><font color="Green">E8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:10_1_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E4:10_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="rfunct_3.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:10_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:10_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="hahnban.html#E3:10_1_1_1"><i><font color="Green">E9</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:10_1_1_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E4:10_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="rfunct_3.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:10_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:10_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="hahnban.html#E3:10_1_1_2"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:10_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : verum </span>}</span>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="hahnban.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E6:10_1_1"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="hahnban.html#E3:10_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="hahnban.html#E4:10_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/><b>hence </b><a NAME="E7:10_1_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="rfunct_3.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2"><b>suppose </b></a><a NAME="E1:10_1_2"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:10_1_2"/>
 <a href="xboole_0.html#K1">{}</a>  is   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="partfun1.html#T56">PARTFUN1:56</a></i>;<br/><b>hence </b><a NAME="E3:10_1_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="rfunct_3.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="hahnban.html#E1:10_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="partfun1.html#T119">PARTFUN1:119</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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