<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> holds <br/> <a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">q</font></span>)</span>,<font color="Maroon" title="c1">X</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Olive" title="b2">q</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font> be   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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"> (  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> implies  <a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">q</font></span>)</span>,<font color="Maroon" title="c1">X</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> )</span><br/>







<span class="kw">assume </span><a NAME="E1:64"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">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">  <a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">q</font></span>)</span>,<font color="Maroon" title="c1">X</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>] <span class="kw">means </span> <a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> $1</span>)</span>,<font color="Maroon" title="c1">X</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> ($1,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>;<br/>
<a NAME="E2:64"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] holds <br/> for <font color="Olive" title="b2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">q</font> be   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font>] implies  for <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 <a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c1">X</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">X</font>)</span>)</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">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">f</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>]</span><br/>

<span class="kw">thus </span> <a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">f</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c1">X</font>) = 
 <a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">f</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c1">X</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c1">X</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T41" target="_self" onmouseover="rs('funct_7/T41')" onmouseout="rh()">Th40</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c6">f</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T41" target="_self" onmouseover="rs('funct_7/T41')" onmouseout="rh()">Th40</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 href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span>)</span>,<font color="Maroon" title="c1">X</font>) = 
 <a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">X</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="relat_1.html#T53" onmouseover="rs('relat_1/T53')" onmouseout="rh()">RELAT_1:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T39" target="_self" onmouseover="rs('funct_7/T39')" onmouseout="rh()">Th38</a></span>
;<br/>
<a NAME="E4:64"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
;<br/>
<a NAME="E5:64"/>
 for <font color="Olive" title="b1">q</font> being   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="funct_7.html#S5" target="_self" onmouseover="rs('funct_7/S5')" onmouseout="rh()">FUNCT_7:sch 5</a>(<a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E28">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:64"/>
 <a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">q</font></span>)</span>,<font color="Maroon" title="c1">X</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K3" title="FUNCT_7:func.3">compose</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">X</font>)</span>)</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>
