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<span class="kw">let </span><font color="Maroon" title="c1">n0</font>, <font color="Maroon" title="c2">m0</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" 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_relative_prime</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="binop_2.html#K24" title="BINOP_2:func.24">*</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="binop_2.html#K24" title="BINOP_2:func.24">*</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="E1:54"/><span class="lab"><font color="Green" title="E50">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_relative_prime</a> 
 ; <a class="txt" 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="binop_2.html#K24" title="BINOP_2:func.24">*</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="binop_2.html#K24" title="BINOP_2:func.24">*</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="E2:54"/><span class="lab"><font color="Green" title="E51">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" title="NAT_5:th.36">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="binop_2.html#K24" title="BINOP_2:func.24">*</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="binop_2.html#K24" title="BINOP_2:func.24">*</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" title="NAT_5:def.3">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="binop_2.html#K24" title="BINOP_2:func.24">*</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="#E1:54"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="nat_5.html#D5" target="_self" title="NAT_5:def.5">Def5</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="binop_2.html#K24" title="BINOP_2:func.24">*</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" title="NAT_5:def.3">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="binop_2.html#K24" title="BINOP_2:func.24">*</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" title="NAT_5:def.3">Def3</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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