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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<br/>  for <font color="Olive" title="b2">ntv</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()  st <font color="Olive" title="b2">ntv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $4 holds <br/><font color="Olive" title="b2">ntv</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>($1,$2,$3,<font color="Olive" title="b1">a</font>);<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">tv</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()  st <font color="Olive" title="b1">tv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 holds <br/> for <font color="Olive" title="b2">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Olive" title="b1">tv</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>($1,<font color="Olive" title="b2">a</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c1">FAB</font> =  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()) as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c2">f29</font> = <font color="Maroon">F<sub>5</sub></font>() as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon" title="c1">FAB</font> ;<br/>
<div><span class="lab"><font color="Green" title="E10">A3</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <font color="Maroon">F<sub>1</sub></font>()  ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <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">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <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">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() = <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c3">x</font>,$1);<br/><span class="kw">consider </span><font color="Maroon" title="c4">y</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()<span class="kw"> such that </span><br/><a NAME="E2:23_1_1"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Maroon" title="c4">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" onmouseover="rs('funct_2/S4')" onmouseout="rh()">FUNCT_2:sch 4</a>();<br/></span><a NAME="E3:23_1_1"/>
( <font color="Maroon">F<sub>2</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="c4">y</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><span class="kw">then </span><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 <font color="Maroon" title="c1">FAB</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">y</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>2</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">y</font>]</span><br/><span class="kw">thus </span><a NAME="E5:23_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1"><span class="lab"><font color="Green" title="E11">A4</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="c3">TV</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon" title="c1">FAB</font><span class="kw"> such that </span><br/><a NAME="E5:23_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">TV</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="#E3:23_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>);<br/></span>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</a> of <font color="Maroon">F<sub>1</sub></font>()) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font> = <font color="Maroon" title="c3">TV</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/>
<div><span class="lab"><font color="Green" title="E12">A6</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_2"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dtconstr.html#K7" title="DTCONSTR:func.7">NonTerminals</a> <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>  ex <font color="Olive" title="b4">fab</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>,<font color="Olive" title="b4">fab</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dtconstr.html#K7" title="DTCONSTR:func.7">NonTerminals</a> <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">  for <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>  ex <font color="Olive" title="b3">fab</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font>,<font color="Olive" title="b3">fab</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</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">fab</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font>,<font color="Olive" title="b1">fab</font>]</span><br/><span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <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">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>3</sub></font>() = <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font>,$1);<br/><span class="kw">consider </span><font color="Maroon" title="c7">fab</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()<span class="kw"> such that </span><br/><a NAME="E2:23_1_2"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Maroon" title="c7">fab</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" onmouseover="rs('funct_2/S4')" onmouseout="rh()">FUNCT_2:sch 4</a>();<br/></span><a NAME="E3:23_1_2"/>
( <font color="Maroon">F<sub>2</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="c7">fab</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">fab</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">fab</font> = <font color="Maroon" title="c7">fab</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c9">fab</font> = <font color="Maroon" title="c8">fab</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="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c9">fab</font>]</span><br/><span class="kw">thus </span><a NAME="E5:23_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c9">fab</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_2"><span class="lab"><font color="Green" title="E13">A7</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="c4">NTV</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"><span class="p2">(<span class="default"><a href="dtconstr.html#K7" title="DTCONSTR:func.7">NonTerminals</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon" title="c1">FAB</font>,<font color="Maroon" title="c1">FAB</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>,<font color="Maroon" title="c1">FAB</font><span class="kw"> such that </span><br/><a NAME="E8:23_1"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dtconstr.html#K7" title="DTCONSTR:func.7">NonTerminals</a> <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>,<font color="Maroon" title="c4">NTV</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="multop_1.html#S1" onmouseover="rs('multop_1/S1')" onmouseout="rh()">MULTOP_1:sch 1</a>(<a class="txt" href="#E6:23_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>);<br/></span>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</a> of <font color="Maroon">F<sub>1</sub></font>(),   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font> = <font color="Maroon" title="c4">NTV</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default">$1,$4,$5</span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E14">A9</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_3"><span class="kw">now </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"> ( (  for <font color="Olive" title="b1">t</font> being   <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">t</font>) ) &amp; (  for <font color="Olive" title="b1">nt</font> being   <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">tl</font>, <font color="Olive" title="b3">tr</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b4">rtl</font>, <font color="Olive" title="b5">rtr</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b4">rtl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b2">tl</font> &amp; <font color="Olive" title="b5">rtr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b3">tr</font> &amp; <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Olive" title="b4">rtl</font>,<font color="Olive" title="b5">rtr</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> holds <br/> for <font color="Olive" title="b6">xl</font>, <font color="Olive" title="b7">xr</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>  st <font color="Olive" title="b6">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">tl</font> &amp; <font color="Olive" title="b7">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">tr</font> holds <br/><font color="Maroon" title="c2">f29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Olive" title="b2">tl</font>,<font color="Olive" title="b3">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">nt</font>,<font color="Olive" title="b4">rtl</font>,<font color="Olive" title="b5">rtr</font>,<font color="Olive" title="b6">xl</font>,<font color="Olive" title="b7">xr</font>) ) )</span></a><div class="add"><div><span class="kw">hereby </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">  for <font color="Olive" title="b1">nt</font> being   <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">tl</font>, <font color="Olive" title="b3">tr</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b4">rtl</font>, <font color="Olive" title="b5">rtr</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b4">rtl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b2">tl</font> &amp; <font color="Olive" title="b5">rtr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b3">tr</font> &amp; <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Olive" title="b4">rtl</font>,<font color="Olive" title="b5">rtr</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> holds <br/> for <font color="Olive" title="b6">xl</font>, <font color="Olive" title="b7">xr</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>  st <font color="Olive" title="b6">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">tl</font> &amp; <font color="Olive" title="b7">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">tr</font> holds <br/><font color="Maroon" title="c2">f29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Olive" title="b2">tl</font>,<font color="Olive" title="b3">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">nt</font>,<font color="Olive" title="b4">rtl</font>,<font color="Olive" title="b5">rtr</font>,<font color="Olive" title="b6">xl</font>,<font color="Olive" title="b7">xr</font>)</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c5">t</font> be   <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</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" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c5">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">t</font>)</span><br/><span class="kw">consider </span><font color="Maroon" title="c6">TVt</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:23_1_3_1"/><span class="lab"><font color="Green" title="E15">A10</font></span>: 
<font color="Maroon" title="c3">TV</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">TVt</font>
 <span class="kw">and </span><br/><a NAME="E3:23_1_3_1"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">TVt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <span class="kw">and </span><br/><a NAME="E4:23_1_3_1"/><span class="lab"><font color="Green" title="E17">A12</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c6">TVt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">TVt</font> = <font color="Maroon" title="c6">TVt</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_3_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E4:23_1_3_1"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" onmouseover="rs('relset_1/T4')" onmouseout="rh()">RELSET_1:4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()<span class="kw"> such that </span><br/><a NAME="E7:23_1_3_1"/><span class="lab"><font color="Green" title="E18">A13</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c5">t</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:23_1_3_1"/><span class="lab"><font color="Green" title="E19">A14</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c5">t</font>,<font color="Olive" title="b1">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E9">A2</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_3_1_1"><span class="kw">now </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">F<sub>2</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="c8">g</font> &amp; <font color="Maroon">F<sub>2</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="c7">TVt</font> &amp; (  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> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">TVt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:23_1_3_1_1"/>
<font color="Maroon">F<sub>2</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="c8">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 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"> ( <font color="Maroon">F<sub>2</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="c7">TVt</font> &amp; (  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> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">TVt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/><span class="kw">thus </span><a NAME="E2:23_1_3_1_1"/>
<font color="Maroon">F<sub>2</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="c7">TVt</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_3_1"><span class="lab"><font color="Green" title="E16">A11</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">  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> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">TVt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() implies <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">TVt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )</span><br/><span class="kw">assume </span><a NAME="E3:23_1_3_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</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" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">TVt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">xx</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/><font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">xx</font> = 
<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c5">t</font>,<font color="Maroon" title="c10">xx</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_3_1"><span class="lab"><font color="Green" title="E19">A14</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">TVt</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">xx</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E2:23_1_3_1"><span class="lab"><font color="Green" title="E15">A10</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E6:23_1_3_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">TVt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</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"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E10:23_1_3_1"/>
<font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c5">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">t</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1_3_1"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="txt" href="#E2:23_1_3_1"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</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">let </span><font color="Maroon" title="c5">nt</font> be   <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</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">  for <font color="Olive" title="b1">tl</font>, <font color="Olive" title="b2">tr</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b3">rtl</font>, <font color="Olive" title="b4">rtr</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b3">rtl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b1">tl</font> &amp; <font color="Olive" title="b4">rtr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b2">tr</font> &amp; <font color="Maroon" title="c5">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Olive" title="b3">rtl</font>,<font color="Olive" title="b4">rtr</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> holds <br/> for <font color="Olive" title="b5">xl</font>, <font color="Olive" title="b6">xr</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>  st <font color="Olive" title="b5">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">tl</font> &amp; <font color="Olive" title="b6">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">tr</font> holds <br/><font color="Maroon" title="c2">f29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Olive" title="b1">tl</font>,<font color="Olive" title="b2">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">nt</font>,<font color="Olive" title="b3">rtl</font>,<font color="Olive" title="b4">rtr</font>,<font color="Olive" title="b5">xl</font>,<font color="Olive" title="b6">xr</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">tl</font>, <font color="Maroon" title="c7">tr</font> be    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <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">  for <font color="Olive" title="b1">rtl</font>, <font color="Olive" title="b2">rtr</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">rtl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c6">tl</font> &amp; <font color="Olive" title="b2">rtr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c7">tr</font> &amp; <font color="Maroon" title="c5">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Olive" title="b1">rtl</font>,<font color="Olive" title="b2">rtr</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> holds <br/> for <font color="Olive" title="b3">xl</font>, <font color="Olive" title="b4">xr</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>  st <font color="Olive" title="b3">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">tl</font> &amp; <font color="Olive" title="b4">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">tr</font> holds <br/><font color="Maroon" title="c2">f29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c6">tl</font>,<font color="Maroon" title="c7">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">nt</font>,<font color="Olive" title="b1">rtl</font>,<font color="Olive" title="b2">rtr</font>,<font color="Olive" title="b3">xl</font>,<font color="Olive" title="b4">xr</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c8">rtl</font>, <font color="Maroon" title="c9">rtr</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</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" title="c8">rtl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c6">tl</font> &amp; <font color="Maroon" title="c9">rtr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c7">tr</font> &amp; <font color="Maroon" title="c5">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c8">rtl</font>,<font color="Maroon" title="c9">rtr</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> implies  for <font color="Olive" title="b1">xl</font>, <font color="Olive" title="b2">xr</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>  st <font color="Olive" title="b1">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">tl</font> &amp; <font color="Olive" title="b2">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">tr</font> holds <br/><font color="Maroon" title="c2">f29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c6">tl</font>,<font color="Maroon" title="c7">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">nt</font>,<font color="Maroon" title="c8">rtl</font>,<font color="Maroon" title="c9">rtr</font>,<font color="Olive" title="b1">xl</font>,<font color="Olive" title="b2">xr</font>) )</span><br/><span class="kw">assume </span><a NAME="E2:23_1_3"/>
( <font color="Maroon" title="c8">rtl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c6">tl</font> &amp; <font color="Maroon" title="c9">rtr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c7">tr</font> &amp; <font color="Maroon" title="c5">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c8">rtl</font>,<font color="Maroon" title="c9">rtr</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</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">  for <font color="Olive" title="b1">xl</font>, <font color="Olive" title="b2">xr</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>  st <font color="Olive" title="b1">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">tl</font> &amp; <font color="Olive" title="b2">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">tr</font> holds <br/><font color="Maroon" title="c2">f29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c6">tl</font>,<font color="Maroon" title="c7">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">nt</font>,<font color="Maroon" title="c8">rtl</font>,<font color="Maroon" title="c9">rtr</font>,<font color="Olive" title="b1">xl</font>,<font color="Olive" title="b2">xr</font>)</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c11">ff1</font>, <font color="Maroon" title="c12">ff2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()<span class="kw"> such that </span><br/><a NAME="E4:23_1_3"/><span class="lab"><font color="Green" title="E15">A15</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c6">tl</font>,<font color="Maroon" title="c7">tr</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:23_1_3"/><span class="lab"><font color="Green" title="E16">A16</font></span>: 
( <font color="Maroon" title="c11">ff1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">tl</font> &amp; <font color="Maroon" title="c12">ff2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">tr</font> &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">nt</font>,<font color="Maroon" title="c11">ff1</font>,<font color="Maroon" title="c12">ff2</font>,<font color="Olive" title="b1">a</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E9">A2</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c13">xl</font>, <font color="Maroon" title="c14">xr</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</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="c13">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">tl</font> &amp; <font color="Maroon" title="c14">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">tr</font> implies <font color="Maroon" title="c2">f29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c6">tl</font>,<font color="Maroon" title="c7">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">nt</font>,<font color="Maroon" title="c8">rtl</font>,<font color="Maroon" title="c9">rtr</font>,<font color="Maroon" title="c13">xl</font>,<font color="Maroon" title="c14">xr</font>) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c15">ntvnt</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E7:23_1_3"/><span class="lab"><font color="Green" title="E17">A17</font></span>: 
<font color="Maroon" title="c15">ntvnt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">NTV</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c5">nt</font>,<font color="Maroon" title="c13">xl</font>,<font color="Maroon" title="c14">xr</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:23_1_3"/><span class="lab"><font color="Green" title="E18">A18</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c15">ntvnt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c15">ntvnt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>3</sub></font>() )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">ntvnt</font> = <font color="Maroon" title="c15">ntvnt</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_3"><span class="lab"><font color="Green" title="E18">A18</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" onmouseover="rs('relset_1/T4')" onmouseout="rh()">RELSET_1:4</a></span>;<br/><span class="kw">assume </span><a NAME="E10:23_1_3"/><span class="lab"><font color="Green" title="E19">A19</font></span>: 
( <font color="Maroon" title="c13">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">tl</font> &amp; <font color="Maroon" title="c14">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">tr</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="c2">f29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c6">tl</font>,<font color="Maroon" title="c7">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">nt</font>,<font color="Maroon" title="c8">rtl</font>,<font color="Maroon" title="c9">rtr</font>,<font color="Maroon" title="c13">xl</font>,<font color="Maroon" title="c14">xr</font>)</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_3_2"><span class="kw">now </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">F<sub>2</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="c10">g</font> &amp; <font color="Maroon">F<sub>2</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="c16">ntvnt</font> &amp; (  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> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ntvnt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:23_1_3_2"/>
<font color="Maroon">F<sub>2</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="c10">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 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"> ( <font color="Maroon">F<sub>2</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="c16">ntvnt</font> &amp; (  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> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ntvnt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/><span class="kw">thus </span><a NAME="E2:23_1_3_2"/>
<font color="Maroon">F<sub>2</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="c16">ntvnt</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 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">  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> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ntvnt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c17">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="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() implies <font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ntvnt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> )</span><br/><span class="kw">assume </span><a NAME="E3:23_1_3_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</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" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ntvnt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">xx</font> = <font color="Maroon" title="c17">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/><font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">xx</font> = 
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c5">nt</font>,<font color="Maroon" title="c13">xl</font>,<font color="Maroon" title="c14">xr</font>,<font color="Maroon" title="c18">xx</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:23_1_3"><span class="lab"><font color="Green" title="E19">A19</font></span></a>, <a class="txt" href="#E5:23_1_3"><span class="lab"><font color="Green" title="E16">A16</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c16">ntvnt</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">xx</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E7:23_1_3"><span class="lab"><font color="Green" title="E17">A17</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E6:23_1_3_2"/>
<font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ntvnt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</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"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E12:23_1_3"/>
<font color="Maroon" title="c2">f29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c6">tl</font>,<font color="Maroon" title="c7">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">nt</font>,<font color="Maroon" title="c8">rtl</font>,<font color="Maroon" title="c9">rtr</font>,<font color="Maroon" title="c13">xl</font>,<font color="Maroon" title="c14">xr</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_3"><span class="lab"><font color="Green" title="E15">A15</font></span></a>, <a class="txt" href="#E7:23_1_3"><span class="lab"><font color="Green" title="E17">A17</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</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">reconsider </span><font color="Maroon" title="c5">f19</font> = <font color="Maroon">F<sub>4</sub></font>() as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon" title="c1">FAB</font> ;<br/>
<div><span class="lab"><font color="Green" title="E15">A20</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_4"><span class="kw">now </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"> ( (  for <font color="Olive" title="b1">t</font> being   <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">t</font>) ) &amp; (  for <font color="Olive" title="b1">nt</font> being   <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">tl</font>, <font color="Olive" title="b3">tr</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b4">rtl</font>, <font color="Olive" title="b5">rtr</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b4">rtl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b2">tl</font> &amp; <font color="Olive" title="b5">rtr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b3">tr</font> &amp; <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Olive" title="b4">rtl</font>,<font color="Olive" title="b5">rtr</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> holds <br/> for <font color="Olive" title="b6">xl</font>, <font color="Olive" title="b7">xr</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>  st <font color="Olive" title="b6">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">tl</font> &amp; <font color="Olive" title="b7">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">tr</font> holds <br/><font color="Maroon" title="c5">f19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Olive" title="b2">tl</font>,<font color="Olive" title="b3">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">nt</font>,<font color="Olive" title="b4">rtl</font>,<font color="Olive" title="b5">rtr</font>,<font color="Olive" title="b6">xl</font>,<font color="Olive" title="b7">xr</font>) ) )</span></a><div class="add"><div><span class="kw">hereby </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">  for <font color="Olive" title="b1">nt</font> being   <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">tl</font>, <font color="Olive" title="b3">tr</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b4">rtl</font>, <font color="Olive" title="b5">rtr</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b4">rtl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b2">tl</font> &amp; <font color="Olive" title="b5">rtr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b3">tr</font> &amp; <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Olive" title="b4">rtl</font>,<font color="Olive" title="b5">rtr</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> holds <br/> for <font color="Olive" title="b6">xl</font>, <font color="Olive" title="b7">xr</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>  st <font color="Olive" title="b6">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">tl</font> &amp; <font color="Olive" title="b7">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">tr</font> holds <br/><font color="Maroon" title="c5">f19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Olive" title="b2">tl</font>,<font color="Olive" title="b3">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">nt</font>,<font color="Olive" title="b4">rtl</font>,<font color="Olive" title="b5">rtr</font>,<font color="Olive" title="b6">xl</font>,<font color="Olive" title="b7">xr</font>)</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c6">t</font> be   <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</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" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">t</font>)</span><br/><span class="kw">consider </span><font color="Maroon" title="c7">TVt</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:23_1_4_1"/><span class="lab"><font color="Green" title="E16">A21</font></span>: 
<font color="Maroon" title="c3">TV</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">TVt</font>
 <span class="kw">and </span><br/><a NAME="E3:23_1_4_1"/><span class="lab"><font color="Green" title="E17">A22</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">TVt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <span class="kw">and </span><br/><a NAME="E4:23_1_4_1"/><span class="lab"><font color="Green" title="E18">A23</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">TVt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">TVt</font> = <font color="Maroon" title="c7">TVt</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_4_1"><span class="lab"><font color="Green" title="E17">A22</font></span></a>, <a class="txt" href="#E4:23_1_4_1"><span class="lab"><font color="Green" title="E18">A23</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" onmouseover="rs('relset_1/T4')" onmouseout="rh()">RELSET_1:4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()<span class="kw"> such that </span><br/><a NAME="E7:23_1_4_1"/><span class="lab"><font color="Green" title="E19">A24</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c6">t</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:23_1_4_1"/><span class="lab"><font color="Green" title="E20">A25</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c6">t</font>,<font color="Olive" title="b1">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_4_1_1"><span class="kw">now </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">F<sub>2</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="c9">g</font> &amp; <font color="Maroon">F<sub>2</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="c8">TVt</font> &amp; (  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> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">TVt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:23_1_4_1_1"/>
<font color="Maroon">F<sub>2</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="c9">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 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"> ( <font color="Maroon">F<sub>2</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="c8">TVt</font> &amp; (  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> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">TVt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/><span class="kw">thus </span><a NAME="E2:23_1_4_1_1"/>
<font color="Maroon">F<sub>2</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="c8">TVt</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_4_1"><span class="lab"><font color="Green" title="E17">A22</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">  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> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">TVt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c10">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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() implies <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">TVt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> )</span><br/><span class="kw">assume </span><a NAME="E3:23_1_4_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</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" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">TVt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">xx</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/><font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">xx</font> = 
<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c6">t</font>,<font color="Maroon" title="c11">xx</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_4_1"><span class="lab"><font color="Green" title="E20">A25</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">TVt</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">xx</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E2:23_1_4_1"><span class="lab"><font color="Green" title="E16">A21</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E6:23_1_4_1_1"/>
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">TVt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</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"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E10:23_1_4_1"/>
<font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">t</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23_1_4_1"><span class="lab"><font color="Green" title="E19">A24</font></span></a>, <a class="txt" href="#E2:23_1_4_1"><span class="lab"><font color="Green" title="E16">A21</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</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">let </span><font color="Maroon" title="c6">nt</font> be   <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</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">  for <font color="Olive" title="b1">tl</font>, <font color="Olive" title="b2">tr</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b3">rtl</font>, <font color="Olive" title="b4">rtr</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b3">rtl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b1">tl</font> &amp; <font color="Olive" title="b4">rtr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b2">tr</font> &amp; <font color="Maroon" title="c6">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Olive" title="b3">rtl</font>,<font color="Olive" title="b4">rtr</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> holds <br/> for <font color="Olive" title="b5">xl</font>, <font color="Olive" title="b6">xr</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>  st <font color="Olive" title="b5">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">tl</font> &amp; <font color="Olive" title="b6">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">tr</font> holds <br/><font color="Maroon" title="c5">f19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Olive" title="b1">tl</font>,<font color="Olive" title="b2">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">nt</font>,<font color="Olive" title="b3">rtl</font>,<font color="Olive" title="b4">rtr</font>,<font color="Olive" title="b5">xl</font>,<font color="Olive" title="b6">xr</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c7">tl</font>, <font color="Maroon" title="c8">tr</font> be    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <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">  for <font color="Olive" title="b1">rtl</font>, <font color="Olive" title="b2">rtr</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">rtl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c7">tl</font> &amp; <font color="Olive" title="b2">rtr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c8">tr</font> &amp; <font color="Maroon" title="c6">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Olive" title="b1">rtl</font>,<font color="Olive" title="b2">rtr</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> holds <br/> for <font color="Olive" title="b3">xl</font>, <font color="Olive" title="b4">xr</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>  st <font color="Olive" title="b3">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">tl</font> &amp; <font color="Olive" title="b4">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tr</font> holds <br/><font color="Maroon" title="c5">f19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c7">tl</font>,<font color="Maroon" title="c8">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">nt</font>,<font color="Olive" title="b1">rtl</font>,<font color="Olive" title="b2">rtr</font>,<font color="Olive" title="b3">xl</font>,<font color="Olive" title="b4">xr</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c9">rtl</font>, <font color="Maroon" title="c10">rtr</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</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" title="c9">rtl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c7">tl</font> &amp; <font color="Maroon" title="c10">rtr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c8">tr</font> &amp; <font color="Maroon" title="c6">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c9">rtl</font>,<font color="Maroon" title="c10">rtr</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> implies  for <font color="Olive" title="b1">xl</font>, <font color="Olive" title="b2">xr</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>  st <font color="Olive" title="b1">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">tl</font> &amp; <font color="Olive" title="b2">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tr</font> holds <br/><font color="Maroon" title="c5">f19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c7">tl</font>,<font color="Maroon" title="c8">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">nt</font>,<font color="Maroon" title="c9">rtl</font>,<font color="Maroon" title="c10">rtr</font>,<font color="Olive" title="b1">xl</font>,<font color="Olive" title="b2">xr</font>) )</span><br/><span class="kw">assume </span><a NAME="E2:23_1_4"/>
( <font color="Maroon" title="c9">rtl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c7">tl</font> &amp; <font color="Maroon" title="c10">rtr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c8">tr</font> &amp; <font color="Maroon" title="c6">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c9">rtl</font>,<font color="Maroon" title="c10">rtr</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</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">  for <font color="Olive" title="b1">xl</font>, <font color="Olive" title="b2">xr</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</font>  st <font color="Olive" title="b1">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">tl</font> &amp; <font color="Olive" title="b2">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tr</font> holds <br/><font color="Maroon" title="c5">f19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c7">tl</font>,<font color="Maroon" title="c8">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">nt</font>,<font color="Maroon" title="c9">rtl</font>,<font color="Maroon" title="c10">rtr</font>,<font color="Olive" title="b1">xl</font>,<font color="Olive" title="b2">xr</font>)</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">g</font>, <font color="Maroon" title="c12">ff1</font>, <font color="Maroon" title="c13">ff2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()<span class="kw"> such that </span><br/><a NAME="E4:23_1_4"/><span class="lab"><font color="Green" title="E16">A26</font></span>: 
<font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c7">tl</font>,<font color="Maroon" title="c8">tr</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:23_1_4"/><span class="lab"><font color="Green" title="E17">A27</font></span>: 
( <font color="Maroon" title="c12">ff1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">tl</font> &amp; <font color="Maroon" title="c13">ff2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tr</font> &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c6">nt</font>,<font color="Maroon" title="c12">ff1</font>,<font color="Maroon" title="c13">ff2</font>,<font color="Olive" title="b1">a</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c14">xl</font>, <font color="Maroon" title="c15">xr</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">FAB</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="c14">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">tl</font> &amp; <font color="Maroon" title="c15">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tr</font> implies <font color="Maroon" title="c5">f19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c7">tl</font>,<font color="Maroon" title="c8">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">nt</font>,<font color="Maroon" title="c9">rtl</font>,<font color="Maroon" title="c10">rtr</font>,<font color="Maroon" title="c14">xl</font>,<font color="Maroon" title="c15">xr</font>) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c16">ntvnt</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E7:23_1_4"/><span class="lab"><font color="Green" title="E18">A28</font></span>: 
<font color="Maroon" title="c16">ntvnt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">NTV</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c6">nt</font>,<font color="Maroon" title="c14">xl</font>,<font color="Maroon" title="c15">xr</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:23_1_4"/><span class="lab"><font color="Green" title="E19">A29</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c16">ntvnt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">ntvnt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>3</sub></font>() )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">ntvnt</font> = <font color="Maroon" title="c16">ntvnt</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_4"><span class="lab"><font color="Green" title="E19">A29</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" onmouseover="rs('relset_1/T4')" onmouseout="rh()">RELSET_1:4</a></span>;<br/><span class="kw">assume </span><a NAME="E10:23_1_4"/><span class="lab"><font color="Green" title="E20">A30</font></span>: 
( <font color="Maroon" title="c14">xl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">tl</font> &amp; <font color="Maroon" title="c15">xr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">tr</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">f19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c7">tl</font>,<font color="Maroon" title="c8">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">nt</font>,<font color="Maroon" title="c9">rtl</font>,<font color="Maroon" title="c10">rtr</font>,<font color="Maroon" title="c14">xl</font>,<font color="Maroon" title="c15">xr</font>)</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1_4_2"><span class="kw">now </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">F<sub>2</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="c11">g</font> &amp; <font color="Maroon">F<sub>2</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="c17">ntvnt</font> &amp; (  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> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">ntvnt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:23_1_4_2"/>
<font color="Maroon">F<sub>2</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="c11">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 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"> ( <font color="Maroon">F<sub>2</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="c17">ntvnt</font> &amp; (  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> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">ntvnt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/><span class="kw">thus </span><a NAME="E2:23_1_4_2"/>
<font color="Maroon">F<sub>2</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="c17">ntvnt</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 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">  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> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">ntvnt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c18">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="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() implies <font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">ntvnt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> )</span><br/><span class="kw">assume </span><a NAME="E3:23_1_4_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</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" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">ntvnt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">xx</font> = <font color="Maroon" title="c18">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/><font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">xx</font> = 
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c6">nt</font>,<font color="Maroon" title="c14">xl</font>,<font color="Maroon" title="c15">xr</font>,<font color="Maroon" title="c19">xx</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:23_1_4"><span class="lab"><font color="Green" title="E20">A30</font></span></a>, <a class="txt" href="#E5:23_1_4"><span class="lab"><font color="Green" title="E17">A27</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">ntvnt</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">xx</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E7:23_1_4"><span class="lab"><font color="Green" title="E18">A28</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E6:23_1_4_2"/>
<font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">ntvnt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</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"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E12:23_1_4"/>
<font color="Maroon" title="c5">f19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c7">tl</font>,<font color="Maroon" title="c8">tr</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">nt</font>,<font color="Maroon" title="c9">rtl</font>,<font color="Maroon" title="c10">rtr</font>,<font color="Maroon" title="c14">xl</font>,<font color="Maroon" title="c15">xr</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_4"><span class="lab"><font color="Green" title="E16">A26</font></span></a>, <a class="txt" href="#E7:23_1_4"><span class="lab"><font color="Green" title="E18">A28</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</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>
<a NAME="E12:23_1"/>
<font color="Maroon" title="c5">f19</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">f29</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="bintree1.html#S4" target="_self" onmouseover="rs('bintree1/S4')" onmouseout="rh()">BINTREE1:sch 4</a>(<a class="txt" href="#E11:23_1"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="txt" href="#E9:23_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E13:23_1"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">F<sub>5</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"> verum</span><br/>


</div>
