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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K6">the_axiom_of_extensionality</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:15"/><i><font color="Green">E12</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_colla.html#T9" target="_self">Th9</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>3</sub></font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>thus </b><a NAME="E4:15"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="ordinal1.html#V1">epsilon-transitive</a>
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green">E12</font></i></a>, <a class="ref" href="zf_colla.html#T12" target="_self">Th12</a></i>;<br/>

<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_colla.html#D3" target="_self">ZF_COLLA:def 3</a><br/>

<b>thus </b><a NAME="E5:15"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green">E12</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_colla.html#D2" target="_self">ZF_COLLA:def 2</a><br/>

<div><i><font color="Green">E13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:15_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E2:15_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="zf_colla.html#E1:15_1"><i><font color="Green">E14</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> );<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:15_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="zf_colla.html#T5" target="_self">Th5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:15_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="zf_colla.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E8:15_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><a NAME="E9:15_1"/><b>then </b>
(  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> or  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="zf_colla.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E10:15_1"/><b>then </b><i><font color="Green">E18</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:15_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="zf_colla.html#E2:15_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="zf_colla.html#E5:15_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="zf_colla.html#E7:15_1"><i><font color="Green">E17</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E12:15_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>(<a class="txt" href="zf_colla.html#E10:15_1"><i><font color="Green">E18</font></i></a>);<br/></i><b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E14:15_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E12:15_1"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E15:15_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green">E12</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E17:15_1"/><i><font color="Green">E22</font></i>: 
( ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> ) )
 <b>by </b><i><a class="txt" href="zf_colla.html#E1:15"><i><font color="Green">E11</font></i></a>, <a class="txt" href="zf_colla.html#E14:15_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zf_colla.html#T13" target="_self">Th13</a></i>;<br/><div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><a NAME="E2:15_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:15_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E14:15_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zf_colla.html#E15:15_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="zf_colla.html#E4:15_1_1"><i><font color="Green">E25</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:15_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E14:15_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zf_colla.html#E1:15_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="zf_colla.html#T6" target="_self">Th6</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:15_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E14:15_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zf_colla.html#E4:15_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zf_colla.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E10:15_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> )
 ;<br/><a NAME="E11:15_1_1"/><b>then </b>
(  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>17</sub></font> or  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="zf_colla.html#T4" target="_self">Th4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E13:15_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E7:15_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zf_colla.html#E9:15_1_1"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E14:15_1_1"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E13:15_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/><b>hence </b><a NAME="E15:15_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_colla.html#E12:15_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="zf_colla.html#E1:15_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="zf_colla.html#E4:15_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zf_colla.html#E13:15_1_1"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/><a NAME="E2:15_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:15_1_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E14:15_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zf_colla.html#E15:15_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="zf_colla.html#E4:15_1_2"><i><font color="Green">E25</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:15_1_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E14:15_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zf_colla.html#E1:15_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="zf_colla.html#T6" target="_self">Th6</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:15_1_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E14:15_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zf_colla.html#E4:15_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zf_colla.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E10:15_1_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> )
 ;<br/><a NAME="E11:15_1_2"/><b>then </b>
(  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>17</sub></font> or  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="zf_colla.html#T4" target="_self">Th4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E13:15_1_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_colla.html#K1" target="_self">Collapse</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E7:15_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zf_colla.html#E9:15_1_2"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E14:15_1_2"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="zf_colla.html#E13:15_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/><b>hence </b><a NAME="E15:15_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_colla.html#E12:15_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="zf_colla.html#E1:15_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="zf_colla.html#E4:15_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zf_colla.html#E13:15_1_2"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E20:15_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_colla.html#E17:15_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="zf_colla.html#E18:15_1"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:15"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E8:15"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>thus </b><a NAME="E10:15"/>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:15_2"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E2:15_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="zf_colla.html#E8:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="zf_colla.html#E1:15_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E3:15_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>


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

<b>given </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E11:15"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E12:15"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:15"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E11:15"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E15:15"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="zf_colla.html#E3:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="zf_colla.html#E6:15"><i><font color="Green">E13</font></i></a>, <a class="txt" href="zf_colla.html#E8:15"><i><font color="Green">E14</font></i></a>, <a class="txt" href="zf_colla.html#E14:15"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:15"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="zf_colla.html#E14:15"><i><font color="Green">E16</font></i></a></i>;<br/>


</div>
