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

<span class="kw">let </span><font color="Maroon" title="c1">n0</font>, <font color="Maroon" title="c2">m0</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="c1">n0</font>,<font color="Maroon" title="c2">m0</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  implies  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">m0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c1">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c2">m0</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:53"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<font color="Maroon" title="c1">n0</font>,<font color="Maroon" title="c2">m0</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">  <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">m0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c1">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c2">m0</font></span>)</span></span><br/>

<a NAME="E2:53"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
 <a href="nat_5.html#K4" title="NAT_5:func.4">Sigma</a> 1 is  <a href="nat_5.html#V1" title="NAT_5:attr.1">multiplicative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T36" target="_self" onmouseover="rs('nat_5/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<span class="kw">thus </span> <a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">m0</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="nat_5.html#K4" title="NAT_5:func.4">Sigma</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">m0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#D3" target="_self" onmouseover="rs('nat_5/D3')" onmouseout="rh()">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K4" title="NAT_5:func.4">Sigma</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K4" title="NAT_5:func.4">Sigma</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">m0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E50">A2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="nat_5.html#K3" title="NAT_5:func.3">sigma</a> (1,<font color="Maroon" title="c1">n0</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K4" title="NAT_5:func.4">Sigma</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">m0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#D3" target="_self" onmouseover="rs('nat_5/D3')" onmouseout="rh()">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c1">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="nat_5.html#K5" title="NAT_5:func.5">sigma</a> <font color="Maroon" title="c2">m0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#D3" target="_self" onmouseover="rs('nat_5/D3')" onmouseout="rh()">Def3</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>
