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<span class="kw">let </span><font color="Maroon" title="c1">b</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  iff <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="monoid_0.html#K4" title="MONOID_0:func.4">&lt;NAT,+&gt;</a>  )</span><br/>

<a NAME="E1:111_1_1"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="monoid_0.html#K4" title="MONOID_0:func.4">&lt;NAT,+&gt;</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D27" target="_self" title="MONOID_0:def.27">Def27</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">n1</font>, <font color="Maroon" title="c3">n2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">n1</font>,<font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="monoid_0.html#K4" title="MONOID_0:func.4">&lt;NAT,+&gt;</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">n1</font>,<font color="Maroon" title="c3">n2</font></span><br/><span class="kw">thus </span><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">n1</font>,<font color="Maroon" title="c3">n2</font> = 
<font color="Maroon" title="c2">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c3">n2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D23" title="BINOP_2:def.23">BINOP_2:def 23</a></span>
<br/>.= 
<a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">n1</font>,<font color="Maroon" title="c3">n2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D20" title="BINOP_2:def.20">BINOP_2:def 20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c2">n1</font>,<font color="Maroon" title="c3">n2</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="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="monoid_0.html#K4" title="MONOID_0:func.4">&lt;NAT,+&gt;</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">n1</font>,<font color="Maroon" title="c3">n2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_1"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="monoid_0.html#T26" target="_self" title="MONOID_0:th.26">Th26</a>, <a class="ref" href="gr_cy_1.html#D4" title="GR_CY_1:def.4">GR_CY_1:def 4</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:111_1_1"/>
( <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  iff <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="monoid_0.html#K4" title="MONOID_0:func.4">&lt;NAT,+&gt;</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_1"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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