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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> for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">w1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font>,<font color="Olive" title="b3">w1</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y1</font>,<font color="Olive" title="b3">w1</font>,$2];<br/>
<a NAME="E1:1_1"/><span class="lab"><font color="Green" title="E2">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> holds <br/> ex <font color="Olive" title="b2">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">z</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <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" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> implies  ex <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">z</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</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">  ex <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">z</font>] )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">y1</font>, <font color="Maroon" title="c4">w1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1"/><span class="lab"><font color="Green" title="E3">A3</font></span>: 
( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c3">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon" title="c4">w1</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="E4:1_1_1"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">w1</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T68" onmouseover="rs('mcart_1/T68')" onmouseout="rh()">MCART_1:68</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:1_1_1"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>()
 <span class="kw">and </span><br/><a NAME="E7:1_1_1"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">w1</font>,<font color="Maroon" title="c5">z</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="txt" href="#E3:1_1_1"><span class="lab"><font color="Green" title="E3">A3</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">z</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" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">z</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E8:1_1_1"/>
<font color="Maroon" title="c5">z</font> <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="#E6:1_1_1"><span class="lab"><font color="Green" title="E5">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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">z</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">x2</font>, <font color="Maroon" title="c7">y2</font>, <font color="Maroon" title="c8">w2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <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" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c8">w2</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</a></span> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c8">w2</font>,<font color="Maroon" title="c5">z</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E9:1_1_1"/><span class="lab"><font color="Green" title="E7">A7a</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">[</a><span class="default"><font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c8">w2</font></span><a href="xtuple_0.html#K3" title="XTUPLE_0:func.3">]</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"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c8">w2</font>,<font color="Maroon" title="c5">z</font>]</span><br/>

<a NAME="E10:1_1_1"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E11:1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font> &amp; <font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c8">w2</font>,<font color="Maroon" title="c5">z</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E9:1_1_1"><span class="lab"><font color="Green" title="E7">A7a</font></span></a>, <a class="txt" href="#E4:1_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</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 <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>,<font color="Maroon">F<sub>4</sub></font>()<span class="kw"> such that </span><br/><a NAME="E3:1_1"/><span class="lab"><font color="Green" title="E3">A8</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E1:1_1"><span class="lab"><font color="Green" title="E2">A2</font></span></a>);<br/></span>
<span class="kw">take </span>
<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">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Olive" title="b3">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">w</font>,<font color="Maroon" title="c1">f</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">w</font>)]</span><br/>

<span class="kw">thus </span><a NAME="E4:1_1"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Olive" title="b3">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">w</font>,<font color="Maroon" title="c1">f</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">w</font>)]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1"><span class="lab"><font color="Green" title="E3">A8</font></span></a>, <a class="ref" href="mcart_1.html#T69" onmouseover="rs('mcart_1/T69')" onmouseout="rh()">MCART_1:69</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>
