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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">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span><font color="Maroon">P<sub>1</sub></font>[$1,$2 <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a> ,$2 <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> ];<br/>
<a NAME="E1:5_1"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  ex <font color="Olive" title="b2">u</font> being    <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>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being    <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>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">e</font>,<font color="Olive" title="b1">u</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">ai</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Maroon" title="c3">bi</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>()<span class="kw"> such that </span><br/><a NAME="E2:5_1_1"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c2">ai</font>,<font color="Maroon" title="c3">bi</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E4">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">u</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">ai</font>,<font color="Maroon" title="c3">bi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> 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>2</sub></font>(),<font color="Maroon">F<sub>3</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="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">u</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c4">u</font>]</span><br/>

<span class="kw">thus </span><a NAME="E4:5_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">e</font>,<font color="Maroon" title="c4">u</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5_1_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><span class="kw"> such that </span><br/><a NAME="E3:5_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">e</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E1:5_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>);<br/></span>
<span class="kw">take </span>
 <a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c1">f</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() st <br/> for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b1">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font>]</span><br/>

<span class="kw">take </span>
 <a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c1">f</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font>]</span><br/>

<a NAME="E4:5_1"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D6" onmouseover="rs('funct_2/D6')" onmouseout="rh()">FUNCT_2:def 6</a></span>;<br/>
<a NAME="E5:5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D5" onmouseover="rs('funct_2/D5')" onmouseout="rh()">FUNCT_2:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:5_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">i</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E4:5_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
