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

<span class="kw">let </span><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">F</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is  <a href="nat_5.html#V1" title="NAT_5:attr.1">multiplicative</a>  &amp; (  for <font color="Olive" title="b1">n0</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Olive" title="b1">n0</font></span>)</span></span>)</span> ) implies <font color="Maroon" title="c2">F</font> is  <a href="nat_5.html#V1" title="NAT_5:attr.1">multiplicative</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:50"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon" title="c1">f</font> is  <a href="nat_5.html#V1" title="NAT_5:attr.1">multiplicative</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"> (  ex <font color="Olive" title="b1">n0</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st  not <font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Olive" title="b1">n0</font></span>)</span></span>)</span> or <font color="Maroon" title="c2">F</font> is  <a href="nat_5.html#V1" title="NAT_5:attr.1">multiplicative</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:50"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
 for <font color="Olive" title="b1">n0</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Olive" title="b1">n0</font></span>)</span></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"> <font color="Maroon" title="c2">F</font> is  <a href="nat_5.html#V1" title="NAT_5:attr.1">multiplicative</a> </span><br/>

<a NAME="E3:50"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">m</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  holds <br/><font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c3">n</font>,<font color="Maroon" title="c4">m</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  implies <font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">b1</font> = <font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">b2</font> = <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">f1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E2:50_1"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c7">f1</font>
 <span class="kw">and </span><br/><a NAME="E3:50_1"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D3" onmouseover="rs('uproots/D3')" onmouseout="rh()">UPROOTS:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">h9</font> = <a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">g2</font> =  <a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">g1</font> =  <a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">f2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E5:50_1"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c11">f2</font>
 <span class="kw">and </span><br/><a NAME="E6:50_1"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c11">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D3" onmouseover="rs('uproots/D3')" onmouseout="rh()">UPROOTS:def 3</a></span>;<br/>
<a NAME="E7:50_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<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/>
<a NAME="E8:50_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A7</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:50_1"/><span class="lab"><font color="Green" title="E53">A8</font></span>: 
<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">m</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</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="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span></span><br/>

<a NAME="E10:50_1"/>
 for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c13">x2</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="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> &amp; <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font> implies <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_1_1"/><span class="lab"><font color="Green" title="E54">A9</font></span>: 
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></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"> (  not <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> or  not <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font> or <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">n1</font>, <font color="Maroon" title="c15">m1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:50_1_1"/><span class="lab"><font color="Green" title="E55">A10</font></span>: 
( <font color="Maroon" title="c14">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c15">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font> )
 <span class="kw">and </span><br/><a NAME="E4:50_1_1"/><span class="lab"><font color="Green" title="E56">A11</font></span>: 
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">n1</font>,<font color="Maroon" title="c15">m1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">n1</font> = <font color="Maroon" title="c14">n1</font>, <font color="Maroon" title="c17">m1</font> = <font color="Maroon" title="c15">m1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_1"><span class="lab"><font color="Green" title="E55">A10</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E57">A12</font></span>: <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> = 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c16">n1</font>,<font color="Maroon" title="c17">m1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c16">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">m1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D24" onmouseover="rs('binop_2/D24')" onmouseout="rh()">BINOP_2:def 24</a></span>
;<br/>
<span class="kw">assume </span><a NAME="E7:50_1_1"/><span class="lab"><font color="Green" title="E58">A13</font></span>: 
<font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></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"> (  not <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font> or <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">n2</font>, <font color="Maroon" title="c19">m2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E9:50_1_1"/><span class="lab"><font color="Green" title="E59">A14</font></span>: 
<font color="Maroon" title="c18">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">and </span><br/><a NAME="E10:50_1_1"/><span class="lab"><font color="Green" title="E60">A15</font></span>: 
<font color="Maroon" title="c19">m2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">and </span><br/><a NAME="E11:50_1_1"/><span class="lab"><font color="Green" title="E61">A16</font></span>: 
<font color="Maroon" title="c13">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">n2</font>,<font color="Maroon" title="c19">m2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">n2</font> = <font color="Maroon" title="c18">n2</font>, <font color="Maroon" title="c21">m2</font> = <font color="Maroon" title="c19">m2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_1_1"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="txt" href="#E10:50_1_1"><span class="lab"><font color="Green" title="E60">A15</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E13:50_1_1"/><span class="lab"><font color="Green" title="E62">A17</font></span>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</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="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2</font></span><br/>

<span class="lab"><font color="Green" title="E63">A18</font></span>: <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font> = 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_1_1"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c20">n2</font>,<font color="Maroon" title="c21">m2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:50_1_1"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c20">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c21">m2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D24" onmouseover="rs('binop_2/D24')" onmouseout="rh()">BINOP_2:def 24</a></span>
;<br/>
<a NAME="E15:50_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">n2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E3:50_1_1"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E9:50_1_1"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="txt" href="#E10:50_1_1"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="txt" href="#E13:50_1_1"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="txt" href="#E6:50_1_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="ref" href="nat_5.html#T15" target="_self" onmouseover="rs('nat_5/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:50_1_1"/>
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E3:50_1_1"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E4:50_1_1"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="txt" href="#E10:50_1_1"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="txt" href="#E11:50_1_1"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="txt" href="#E13:50_1_1"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="txt" href="#E6:50_1_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E14:50_1_1"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="ref" href="nat_5.html#T15" target="_self" onmouseover="rs('nat_5/T15')" onmouseout="rh()">Th15</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">h9</font> = <a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> as   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">h</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E12:50_1"/><span class="lab"><font color="Green" title="E54">A19</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T37" onmouseover="rs('pre_poly/T37')" onmouseout="rh()">PRE_POLY:37</a></span>;<br/>
<a NAME="E13:50_1"/><span class="lab"><font color="Green" title="E55">A20</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<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/>
<a NAME="E14:50_1"/>
 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>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> implies <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> )</span><br/>

<a NAME="E1:50_1_2"/><span class="lab"><font color="Green" title="E56">A21</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:50_1_2"/><span class="lab"><font color="Green" title="E57">A22</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></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"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span></span><br/>

<a NAME="E3:50_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A23</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>;<br/>
<a NAME="E4:50_1_2"/><span class="lab"><font color="Green" title="E59">A24</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_2"><span class="lab"><font color="Green" title="E57">A22</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">n1</font>, <font color="Maroon" title="c16">m1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:50_1_2"/><span class="lab"><font color="Green" title="E60">A25</font></span>: 
( <font color="Maroon" title="c15">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c16">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font> )
 <span class="kw">and </span><br/><a NAME="E7:50_1_2"/><span class="lab"><font color="Green" title="E61">A26</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">n1</font>,<font color="Maroon" title="c16">m1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_2"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">n1</font> = <font color="Maroon" title="c15">n1</font>, <font color="Maroon" title="c18">m1</font> = <font color="Maroon" title="c16">m1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1_2"><span class="lab"><font color="Green" title="E60">A25</font></span></a></span>;<br/>
<a NAME="E9:50_1_2"/><span class="lab"><font color="Green" title="E62">A27</font></span>: 
<font color="Maroon" title="c17">n1</font>,<font color="Maroon" title="c18">m1</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E6:50_1_2"><span class="lab"><font color="Green" title="E60">A25</font></span></a>, <a class="ref" href="nat_5.html#T14" target="_self" onmouseover="rs('nat_5/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">n19</font> = <font color="Maroon" title="c17">n1</font>, <font color="Maroon" title="c20">m19</font> = <font color="Maroon" title="c18">m1</font> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1_2"><span class="lab"><font color="Green" title="E60">A25</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">x9</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">m1</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">fn1</font> = <font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">n1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">fm1</font> = <font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">m1</font>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_2"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:50_1"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E4:50_1_2"><span class="lab"><font color="Green" title="E59">A24</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c17">n1</font>,<font color="Maroon" title="c18">m1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_1_2"><span class="lab"><font color="Green" title="E61">A26</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">m1</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T75" onmouseover="rs('funct_3/T75')" onmouseout="rh()">FUNCT_3:75</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">n1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">m1</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D24" onmouseover="rs('binop_2/D24')" onmouseout="rh()">BINOP_2:def 24</a></span>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">n19</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">m19</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E9:50_1_2"><span class="lab"><font color="Green" title="E62">A27</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1_2"><span class="lab"><font color="Green" title="E60">A25</font></span></a>, <a class="ref" href="nat_5.html#T16" target="_self" onmouseover="rs('nat_5/T16')" onmouseout="rh()">Th16</a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<a NAME="E12:50_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A28</font></span>: 
<font color="Maroon" title="c17">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">m1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_2"><span class="lab"><font color="Green" title="E58">A23</font></span></a>, <a class="txt" href="#E1:50_1_2"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>;<br/>
<a NAME="E13:50_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">m1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/>
<a NAME="E14:50_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">m1</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E15:50_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A29</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">m1</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">m1</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c17">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">m1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:50_1_2"><span class="lab"><font color="Green" title="E63">A28</font></span></a>, <a class="ref" href="funct_1.html#T35" onmouseover="rs('funct_1/T35')" onmouseout="rh()">FUNCT_1:35</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c17">n1</font>,<font color="Maroon" title="c18">m1</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D24" onmouseover="rs('binop_2/D24')" onmouseout="rh()">BINOP_2:def 24</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">n1</font>,<font color="Maroon" title="c18">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c12">h9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_2"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="txt" href="#E7:50_1_2"><span class="lab"><font color="Green" title="E61">A26</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<a NAME="E17:50_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">h9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:50_1_2"><span class="lab"><font color="Green" title="E64">A29</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E18:50_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">h9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:50_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E2:50_1_2"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</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="E15:50_1"/><span class="kw">then </span>
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span>
 
;<br/>
<a NAME="E16:50_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span>
 
;<br/>
<a NAME="E17:50_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A30</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E18:50_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:50_1"><span class="lab"><font color="Green" title="E54">A19</font></span></a></span>;<br/>
<a NAME="E19:50_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A31</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E17:50_1"><span class="lab"><font color="Green" title="E56">A30</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E20:50_1"/><span class="lab"><font color="Green" title="E58">A32</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">f2</font> holds <br/><font color="Maroon" title="c11">f2</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">f1</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p0">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">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="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">f2</font> implies <font color="Maroon" title="c11">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p0">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c15">x9</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>;<br/>
<span class="lab"><font color="Green" title="E59">A33</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font>
<span class="kw">by</span> <span class="lab"><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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font>
<span class="kw">by</span> <span class="lab"><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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T39" onmouseover="rs('funct_1/T39')" onmouseout="rh()">FUNCT_1:39</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>
;<br/>
<span class="kw">assume </span><a NAME="E2:50_1_3"/><span class="lab"><font color="Green" title="E60">A34</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">f2</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">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p0">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span><br/>

<a NAME="E3:50_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A35</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<a NAME="E4:50_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A36</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E5:50_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A37</font></span>: 
(  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T37" onmouseover="rs('pre_poly/T37')" onmouseout="rh()">PRE_POLY:37</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">n1</font>, <font color="Maroon" title="c17">m1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:50_1_3"/><span class="lab"><font color="Green" title="E64">A38</font></span>: 
( <font color="Maroon" title="c16">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c17">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font> )
 <span class="kw">and </span><br/><a NAME="E8:50_1_3"/><span class="lab"><font color="Green" title="E65">A39</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">n1</font>,<font color="Maroon" title="c17">m1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">n1</font> = <font color="Maroon" title="c16">n1</font>, <font color="Maroon" title="c19">m1</font> = <font color="Maroon" title="c17">m1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_1_3"><span class="lab"><font color="Green" title="E64">A38</font></span></a></span>;<br/>
<a NAME="E10:50_1_3"/><span class="lab"><font color="Green" title="E66">A40</font></span>: 
<font color="Maroon" title="c18">n1</font>,<font color="Maroon" title="c19">m1</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E7:50_1_3"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="ref" href="nat_5.html#T14" target="_self" onmouseover="rs('nat_5/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">n19</font> = <font color="Maroon" title="c18">n1</font>, <font color="Maroon" title="c21">m19</font> = <font color="Maroon" title="c19">m1</font> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_1_3"><span class="lab"><font color="Green" title="E64">A38</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">fn1</font> = <font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">n1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">fm1</font> = <font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">m1</font>;<br/>
<a NAME="E12:50_1_3"/><span class="lab"><font color="Green" title="E67">A41</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_1_3"><span class="lab"><font color="Green" title="E63">A37</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E68">A42</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_1_3"><span class="lab"><font color="Green" title="E63">A37</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></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"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:50_1"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E12:50_1_3"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c18">n1</font>,<font color="Maroon" title="c19">m1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_1_3"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">m1</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T75" onmouseover="rs('funct_3/T75')" onmouseout="rh()">FUNCT_3:75</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">n1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">m1</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D24" onmouseover="rs('binop_2/D24')" onmouseout="rh()">BINOP_2:def 24</a></span>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n19</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c21">m19</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E10:50_1_3"><span class="lab"><font color="Green" title="E66">A40</font></span></a></span>
;<br/>
<a NAME="E14:50_1_3"/><span class="lab"><font color="Green" title="E69">A43</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_3"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>;<br/>
<a NAME="E15:50_1_3"/>
<font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n19</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c21">m19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c19">m1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_1_3"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="ref" href="nat_5.html#T16" target="_self" onmouseover="rs('nat_5/T16')" onmouseout="rh()">Th16</a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>;<br/>
<a NAME="E16:50_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A44</font></span>: 
<font color="Maroon" title="c18">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c19">m1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:50_1_3"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E13:50_1_3"><span class="lab"><font color="Green" title="E68">A42</font></span></a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>;<br/>
<a NAME="E17:50_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A45</font></span>: 
<font color="Maroon" title="c18">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c19">m1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
;<br/>
<span class="lab"><font color="Green" title="E72">A46</font></span>: <font color="Maroon" title="c12">h9</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"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> = 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E5:50_1_3"><span class="lab"><font color="Green" title="E63">A37</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c18">n1</font>,<font color="Maroon" title="c19">m1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_1_3"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c18">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c19">m1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D24" onmouseover="rs('binop_2/D24')" onmouseout="rh()">BINOP_2:def 24</a></span>
;<br/>
<span class="lab"><font color="Green" title="E73">A47</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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"><span class="p5">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p0">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p0">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:50_1"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="txt" href="#E2:50_1_3"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<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="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p0">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
<span class="kw">by</span> <span class="lab"><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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_3"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</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="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E5:50_1_3"><span class="lab"><font color="Green" title="E63">A37</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c19">m1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:50_1_3"><span class="lab"><font color="Green" title="E71">A45</font></span></a>, <a class="txt" href="#E18:50_1_3"><span class="lab"><font color="Green" title="E72">A46</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c19">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:50_1_3"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_1_3"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="txt" href="#E13:50_1_3"><span class="lab"><font color="Green" title="E68">A42</font></span></a>, <a class="ref" href="nat_5.html#T16" target="_self" onmouseover="rs('nat_5/T16')" onmouseout="rh()">Th16</a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E21:50_1_3"/>
<font color="Maroon" title="c11">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p0">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E6:50_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E3:50_1_3"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="txt" href="#E19:50_1_3"><span class="lab"><font color="Green" title="E73">A47</font></span></a>, <a class="txt" href="#E1:50_1_3"><span class="lab"><font color="Green" title="E59">A33</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</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="E21:50_1"/>
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">y</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="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> implies <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_1_4"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></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"> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E2:50_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A48</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">n1</font>, <font color="Maroon" title="c16">m1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:50_1_4"/><span class="lab"><font color="Green" title="E60">A49</font></span>: 
( <font color="Maroon" title="c15">n1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c16">m1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font> )
 <span class="kw">and </span><br/><a NAME="E5:50_1_4"/><span class="lab"><font color="Green" title="E61">A50</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E2:50_1_4"><span class="lab"><font color="Green" title="E59">A48</font></span></a>, <a class="ref" href="nat_5.html#T18" target="_self" onmouseover="rs('nat_5/T18')" onmouseout="rh()">Th18</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">n19</font> = <font color="Maroon" title="c15">n1</font>, <font color="Maroon" title="c18">m19</font> = <font color="Maroon" title="c16">m1</font> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_4"><span class="lab"><font color="Green" title="E60">A49</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">n1</font> = <font color="Maroon" title="c15">n1</font>, <font color="Maroon" title="c20">m1</font> = <font color="Maroon" title="c16">m1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E8:50_1_4"/><span class="lab"><font color="Green" title="E62">A51</font></span>: 
<font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E4:50_1_4"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="ref" href="nat_5.html#T14" target="_self" onmouseover="rs('nat_5/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<a NAME="E9:50_1_4"/><span class="lab"><font color="Green" title="E63">A52</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_4"><span class="lab"><font color="Green" title="E59">A48</font></span></a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>;<br/>
<a NAME="E10:50_1_4"/><span class="lab"><font color="Green" title="E64">A53</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_4"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">x</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<a NAME="E11:50_1_4"/><span class="lab"><font color="Green" title="E65">A54</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_4"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> = 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_4"><span class="lab"><font color="Green" title="E59">A48</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">n19</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">m19</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E5:50_1_4"><span class="lab"><font color="Green" title="E61">A50</font></span></a>, <a class="txt" href="#E8:50_1_4"><span class="lab"><font color="Green" title="E62">A51</font></span></a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">n1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">m1</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D24" onmouseover="rs('binop_2/D24')" onmouseout="rh()">BINOP_2:def 24</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">m1</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T75" onmouseover="rs('funct_3/T75')" onmouseout="rh()">FUNCT_3:75</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:50_1"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:50_1_4"><span class="lab"><font color="Green" title="E65">A54</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<a NAME="E13:50_1_4"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_1_4"><span class="lab"><font color="Green" title="E63">A52</font></span></a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>;<br/>
<a NAME="E14:50_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A55</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E15:50_1_4"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/>
<a NAME="E16:50_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E17:50_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A56</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/>
<span class="lab"><font color="Green" title="E68">A57</font></span>: <font color="Maroon" title="c14">y</font> = 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_1_4"><span class="lab"><font color="Green" title="E61">A50</font></span></a>, <a class="ref" href="binop_2.html#D24" onmouseover="rs('binop_2/D24')" onmouseout="rh()">BINOP_2:def 24</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:50_1_4"><span class="lab"><font color="Green" title="E64">A53</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c12">h9</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"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></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"><span class="p4">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:50_1_4"><span class="lab"><font color="Green" title="E66">A55</font></span></a>, <a class="ref" href="funct_1.html#T35" onmouseover="rs('funct_1/T35')" onmouseout="rh()">FUNCT_1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></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"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:50_1_4"><span class="lab"><font color="Green" title="E67">A56</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
;<br/>
<a NAME="E19:50_1_4"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></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"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:50_1_4"><span class="lab"><font color="Green" title="E66">A55</font></span></a>, <a class="ref" href="funct_1.html#T35" onmouseover="rs('funct_1/T35')" onmouseout="rh()">FUNCT_1:35</a></span>;<br/>
<a NAME="E20:50_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">n1</font>,<font color="Maroon" title="c20">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E10:50_1_4"><span class="lab"><font color="Green" title="E64">A53</font></span></a>, <a class="txt" href="#E17:50_1_4"><span class="lab"><font color="Green" title="E67">A56</font></span></a>, <a class="ref" href="funct_1.html#T11" onmouseover="rs('funct_1/T11')" onmouseout="rh()">FUNCT_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:50_1_4"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:50_1_4"><span class="lab"><font color="Green" title="E68">A57</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E22:50_1"/><span class="kw">then </span>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E23:50_1"/><span class="kw">then </span>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/>
<a NAME="E24:50_1"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T28" onmouseover="rs('relat_1/T28')" onmouseout="rh()">RELAT_1:28</a></span>;<br/>
<a NAME="E25:50_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A58</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p0"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/>
<a NAME="E26:50_1"/>
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T37" onmouseover="rs('pre_poly/T37')" onmouseout="rh()">PRE_POLY:37</a></span>;<br/>
<a NAME="E27:50_1"/><span class="kw">then </span>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E28:50_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E29:50_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A59</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</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"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E25:50_1"><span class="lab"><font color="Green" title="E59">A58</font></span></a>, <a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>;<br/>
<a NAME="E30:50_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">f2</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">f1</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:50_1"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E31:50_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</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"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p5">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">h9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p5"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:50_1"><span class="lab"><font color="Green" title="E58">A32</font></span></a>, <a class="ref" href="funct_1.html#T10" onmouseover="rs('funct_1/T10')" onmouseout="rh()">FUNCT_1:10</a></span>;<br/>
<a NAME="E32:50_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A60</font></span>: 
<font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c11">f2</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:50_1"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="txt" href="#E29:50_1"><span class="lab"><font color="Green" title="E60">A59</font></span></a>, <a class="ref" href="classes1.html#T77" onmouseover="rs('classes1/T77')" onmouseout="rh()">CLASSES1:77</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span> = 
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>
<br/>.= 
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E5:50_1"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E32:50_1"><span class="lab"><font color="Green" title="E61">A60</font></span></a>, <a class="ref" href="rfinseq.html#T9" onmouseover="rs('rfinseq/T9')" onmouseout="rh()">RFINSEQ:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T28" target="_self" onmouseover="rs('nat_5/T28')" onmouseout="rh()">Th28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E47">A2</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">hence </span><a NAME="E4:50"/>
<font color="Maroon" title="c2">F</font> is  <a href="nat_5.html#V1" title="NAT_5:attr.1">multiplicative</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"> verum</span><br/>


</div>
