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<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p1">(<span class="default">$1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c1">M</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">u9</font>,<font color="Olive" title="b2">v9</font>) where <font color="Olive" title="b1">u9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:37_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:37_1"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span>
<a NAME="E4:37_1"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></span>;<br/>
<a NAME="E5:37_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">u9</font>,<font color="Olive" title="b2">v9</font>) where <font color="Olive" title="b1">u9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:37_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">u9</font>,<font color="Olive" title="b2">v9</font>) where <font color="Olive" title="b1">u9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">u9</font>,<font color="Olive" title="b2">v9</font>) where <font color="Olive" title="b1">u9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</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="c3">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">u9</font>,<font color="Olive" title="b2">v9</font>) where <font color="Olive" title="b1">u9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span>  or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_1_1_1"/>
<font color="Maroon" title="c3">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">u9</font>,<font color="Olive" title="b2">v9</font>) where <font color="Olive" title="b1">u9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span> 
 ; <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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c5">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E3:37_1_1_1"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>)
 <span class="kw">and </span><br/><a NAME="E4:37_1_1_1"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">and </span><br/><a NAME="E5:37_1_1_1"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>()
 ;<br/>
<a NAME="E6:37_1_1_1"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1_1_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E5:37_1_1_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E7:37_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A10</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E8:37_1_1_1"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E9:37_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E10:37_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E7:37_1_1_1"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E8:37_1_1_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:37_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_1_1_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="txt" href="#E6:37_1_1_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E7:37_1_1_1"><span class="lab"><font color="Green" title="E15">A10</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">let </span><font color="Maroon" title="c3">x</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> or <font color="Maroon" title="c3">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">u9</font>,<font color="Olive" title="b2">v9</font>) where <font color="Olive" title="b1">u9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:37_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <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="c3">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">u9</font>,<font color="Olive" title="b2">v9</font>) where <font color="Olive" title="b1">u9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:37_1_1"/><span class="lab"><font color="Green" title="E11">A12</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E5:37_1_1"/><span class="lab"><font color="Green" title="E12">A13</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:37_1_1"/><span class="lab"><font color="Green" title="E13">A14</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E11">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E8:37_1_1"/><span class="lab"><font color="Green" title="E14">A15</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></span>;<br/>
<a NAME="E9:37_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A16</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E10:37_1_1"/><span class="lab"><font color="Green" title="E16">A17</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="txt" href="#E8:37_1_1"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E11:37_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E3:37_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E4:37_1_1"><span class="lab"><font color="Green" title="E11">A12</font></span></a>, <a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:37_1_1"/>
<font color="Maroon" title="c3">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">u9</font>,<font color="Olive" title="b2">v9</font>) where <font color="Olive" title="b1">u9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_1_1"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="txt" href="#E10:37_1_1"><span class="lab"><font color="Green" title="E16">A17</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">hence </span><a NAME="E6:37_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">u9</font>,<font color="Olive" title="b2">v9</font>) where <font color="Olive" title="b1">u9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">v9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( <font color="Olive" title="b1">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b2">v9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E7">A2</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>
