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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</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"> for <font color="Olive" title="b1">I</font> being  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>  the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds   <a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <font color="Olive" title="b1">I</font>,<font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <font color="Olive" title="b1">I</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">I</font> be   <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>  the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <font color="Maroon" title="c3">I</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span><br/>





<a NAME="E1:29"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_4.html#D6" target="_self" title="SCMFSA_4:def.6">Def6</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E21">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">l</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"> ( <font color="Maroon" title="c4">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">I</font> implies <span class="p1">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">IncAddr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">l</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:29_1"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c4">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">IncAddr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">l</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span><br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c1">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">l</font> = 
<span class="p1">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c1">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
 <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">IncAddr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">l</font></span>)</span>,<font color="Maroon" title="c1">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="scmfsa_4.html#T24" target="_self" title="SCMFSA_4:th.24">Th24</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">l</font> = 
 <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">IncAddr</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">IncAddr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">l</font></span>)</span>,<font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c2">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E1:29_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="scmfsa_4.html#T24" target="_self" title="SCMFSA_4:th.24">Th24</a></span>
<br/>.= 
 <a href="scmfsa_4.html#K1" title="SCMFSA_4:func.1">IncAddr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">l</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_4.html#T23" target="_self" title="SCMFSA_4:th.23">Th23</a></span>
;<br/> <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>
<a NAME="E3:29"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c1">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_4.html#D6" target="_self" title="SCMFSA_4:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:29"/>
 <a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_4.html#K2" title="SCMFSA_4:func.2">IncAddr</a> <font color="Maroon" title="c3">I</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="scmfsa_4.html#D6" target="_self" title="SCMFSA_4:def.6">Def6</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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