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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">fp</font>, <font color="Olive" title="b4">fq</font> being  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a>  st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Olive" title="b3">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">fq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">q</font> holds <br/><font color="Olive" title="b3">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b4">fq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b2">q</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">fp</font>, <font color="Maroon" title="c5">fq</font> be   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c4">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c5">fq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">q</font> implies <font color="Maroon" title="c4">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">fq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c3">q</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:117"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c4">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c5">fq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">q</font> )
 ; <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="c4">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">fq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">pq</font> = <font color="Maroon" title="c2">p</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c3">q</font>;<br/>
<a NAME="E2:117"/><span class="lab"><font color="Green" title="E68">C2</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T29" title="AFINSQ_1:th.29">AFINSQ_1:29</a></span>;<br/>
<a NAME="E3:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A2</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="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c3">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">fpq</font> = <font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c3">q</font></span>)</span> as   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E68">C2</font></span></a>, <a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a>, <a class="ref" href="afinsq_1.html#T13" title="AFINSQ_1:th.13">AFINSQ_1:13</a></span>;<br/>
<a NAME="E5:117"/><span class="lab"><font color="Green" title="E70">A3</font></span>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">p</font> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">fq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c3">q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<a NAME="E6:117"/><span class="lab"><font color="Green" title="E71">A4</font></span>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font></span>)</span> &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">fq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">fp</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">fq</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>;<br/>
<a NAME="E7:117"/><span class="lab"><font color="Green" title="E72">A5</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">fpq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">fq</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E68">C2</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a>, <a class="txt" href="#E5:117"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="txt" href="#E6:117"><span class="lab"><font color="Green" title="E71">A4</font></span></a></span>;<br/>
<a NAME="E8:117"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">fpq</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">fq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fpq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">fpq</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">fq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fpq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:117_1"/><span class="lab"><font color="Green" title="E73">G1</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">fpq</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"><font color="Maroon" title="c4">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">fq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fpq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span><br/>

<a NAME="E2:117_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">G2</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fpq</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:117_1_1"/>
( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> or <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:117_1_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">p</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"><font color="Maroon" title="c4">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">fq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fpq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span><br/><div class="add"><a NAME="E2:117_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E3:117_1_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">fq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E5:117"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><span class="kw">hence </span><a NAME="E4:117_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">fq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fpq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_1"><span class="lab"><font color="Green" title="E74">G2</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:117_1_1_2"/><span class="lab"><font color="Green" title="E75">G3</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">p</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"><font color="Maroon" title="c4">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">fq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fpq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">kp</font> = <font color="Maroon" title="c8">k</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/><a NAME="E3:117_1_1_2"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c9">kp</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:117"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E68">C2</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a>, <a class="txt" href="#E1:117_1"><span class="lab"><font color="Green" title="E73">G1</font></span></a></span>;<br/><a NAME="E4:117_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">kp</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E5:117_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">kp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><a NAME="E6:117_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">kp</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">fq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">fq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">kp</font> &amp; <font color="Maroon" title="c5">fq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">kp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">kp</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T21" title="AFINSQ_1:th.21">AFINSQ_1:21</a>, <a class="txt" href="#E1:117_1"><span class="lab"><font color="Green" title="E73">G1</font></span></a>, <a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E3:117"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="txt" href="#E5:117"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="txt" href="#E6:117"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="txt" href="#E1:117_1_1_2"><span class="lab"><font color="Green" title="E75">G3</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><span class="kw">hence </span><a NAME="E7:117_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">fq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fpq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_1"><span class="lab"><font color="Green" title="E74">G2</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:117"/>
<font color="Maroon" title="c4">fp</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">fq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T11" title="AFINSQ_1:th.11">AFINSQ_1:11</a>, <a class="txt" href="#E7:117"><span class="lab"><font color="Green" title="E72">A5</font></span></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>
