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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span><br/> for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b4">Q</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c3">C</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Olive" title="b4">Q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">Q</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Olive" title="b2">g</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b2">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">Q</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c3">C</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Olive" title="b3">Q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">Q</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Olive" title="b1">g</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b2">Q</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c3">C</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Olive" title="b2">Q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">g</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">P</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Q</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c3">C</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Olive" title="b1">Q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Q</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">g</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">Q</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c3">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Q</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">g</font></span>)</span></span><br/>









<a NAME="E1:31"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:31"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#T20" target="_self" title="HILBERT3:th.20">Th20</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:31"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:31"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></span>;<br/>
<a NAME="E5:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A4</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<a NAME="E6:31"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">fx</font> = <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></span>;<br/>
<a NAME="E3:31_1"/>
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<a NAME="E4:31_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">fx</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E5:31_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<a NAME="E6:31_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> = 
<font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c7">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">fx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">fx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">fx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">fx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T88" title="FUNCT_2:th.88">FUNCT_2:88</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">fx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">fx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">fx</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">fx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#D1" target="_self" title="HILBERT3:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:31"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">Q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Q</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
