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<span class="kw">defpred </span><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> ] <span class="kw">means </span>for <font color="Olive" title="b1">phi</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()  st <font color="Olive" title="b1">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">phi</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c1">Z</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">phi9</font>) where <font color="Olive" title="b1">phi9</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Olive" title="b1">phi9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>() </span>}</span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">phi9</font>) where <font color="Olive" title="b1">phi9</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Olive" title="b1">phi9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>() </span>}</span> ;<br/>
<a NAME="E2:42_1"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:42_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
not <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">phi9</font>) where <font color="Olive" title="b1">phi9</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Olive" title="b1">phi9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>() </span>}</span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:42_1"/><span class="kw">then </span>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">phi9</font>) where <font color="Olive" title="b1">phi9</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Olive" title="b1">phi9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>() </span>}</span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E5:42_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">phi9</font>) where <font color="Olive" title="b1">phi9</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Olive" title="b1">phi9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>() </span>}</span> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">phi</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E7:42_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c3">phi</font>)
 <span class="kw">and </span><br/><a NAME="E8:42_1"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
<font color="Maroon" title="c3">phi</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_1_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><a NAME="E2:42_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<font color="Maroon" title="c3">phi</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/><a NAME="E3:42_1_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">phi</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E4:42_1_1"/><span class="kw">then </span>
<font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="relat_1.html#T151" title="RELAT_1:th.151">RELAT_1:151</a></span>;<br/><span class="kw">hence </span><a NAME="E5:42_1_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">FF</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span>, <font color="Maroon" title="c5">Z</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">phi9</font>) where <font color="Olive" title="b1">phi9</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() : <font color="Olive" title="b1">phi9</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>() </span>}</span>  as  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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<a NAME="E11:42_1"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E12:42_1"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E15">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">FF</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Z</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">g</font>]</span><br/><span class="kw">consider </span><font color="Maroon" title="c7">phi</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E2:42_1_2"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<font color="Maroon" title="c7">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="fraenkel.html#T14" target="_self" title="FRAENKEL:th.14">Th14</a></span>;<br/><a NAME="E3:42_1_2"/><span class="lab"><font color="Green" title="E17">A12</font></span>: 
<font color="Maroon" title="c7">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="fraenkel.html#T15" target="_self" title="FRAENKEL:th.15">Th15</a></span>;<br/><a NAME="E4:42_1_2"/>
 ex <font color="Olive" title="b1">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>() &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>() )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/><a NAME="E5:42_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E3:42_1_2"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E6:42_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">phi</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></span>;<br/><a NAME="E7:42_1_2"/><span class="kw">then </span>
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c7">phi</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">g9</font> = <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c7">phi</font>) as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Z</font> ;<br/><span class="kw">take </span><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">g9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c9">g</font>]</span><br/><span class="kw">thus </span><a NAME="E9:42_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c9">g</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E3:42_1_2"><span class="lab"><font color="Green" title="E17">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c6">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">FF</font>,<font color="Maroon" title="c5">Z</font><span class="kw"> such that </span><br/><a NAME="E15:42_1"/><span class="lab"><font color="Green" title="E16">A13</font></span>: 
for <font color="Olive" title="b1">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">FF</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E13:42_1"><span class="lab"><font color="Green" title="E15">A10</font></span></a>);<br/></span>
<a NAME="E16:42_1"/>
<font color="Maroon" title="c5">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> or <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_3"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">phi</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E3:42_1_3"/><span class="lab"><font color="Green" title="E17">A14</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c8">phi</font>)
 <span class="kw">and </span><br/><a NAME="E4:42_1_3"/><span class="lab"><font color="Green" title="E18">A15</font></span>: 
<font color="Maroon" title="c8">phi</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 ;<br/>
<a NAME="E5:42_1_3"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>4</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_3"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></span>;<br/>
<a NAME="E6:42_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A16</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<span class="lab"><font color="Green" title="E20">A17</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">phi</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon">F<sub>3</sub></font>()
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
<font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>() as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_3"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E9:42_1_3"/>
<font color="Maroon" title="c8">phi</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_1_3"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E6:42_1_3"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E10:42_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A18</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E11:42_1_3"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42_1"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E3:42_1_3"><span class="lab"><font color="Green" title="E17">A14</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:42_1_3"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_1_3"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E17:42_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
