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<b>let </b><font color="Maroon" title="c1">N</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c1">N</font><br/> for <font color="Olive" title="b2">p</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">s</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b4">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K10" title="SCMNORM:func.10">Following</a> <font color="Olive" title="b2">p</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">s</font>,<font color="Olive" title="b4">k</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">s</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b3">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K10" title="SCMNORM:func.10">Following</a> <font color="Olive" title="b1">p</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">s</font>,<font color="Olive" title="b3">k</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">p</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K10" title="SCMNORM:func.10">Following</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">k</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">s</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds   <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K10" title="SCMNORM:func.10">Following</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<font color="Olive" title="b1">k</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K10" title="SCMNORM:func.10">Following</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c5">k</font></span>)</span></span><br/>









<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Maroon" title="c2">S</font>) <b>-&gt; </b>  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <font color="Maroon" title="c2">S</font> =  <a href="scmnorm.html#K10" title="SCMNORM:func.10">Following</a> <font color="Maroon" title="c3">p</font>,$2;<br/>
<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E2:30"/><i><font color="Green" title="E6">W1</font></i>: 
 <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green" title="E7">W2</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font>
 <b>and </b><br/><a NAME="E4:30"/><i><font color="Green" title="E8">W3</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c6">f</font> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font>)
 <b>by </b><i><a class="ref" href="scmnorm.html#D6" target="_self" title="SCMNORM:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E6:30"/><i><font color="Green" title="E9">V1</font></i>: 
 <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font>
 <b>and </b><br/><a NAME="E7:30"/><i><font color="Green" title="E10">V2</font></i>: 
<font color="Maroon" title="c7">g</font> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font>
 <b>and </b><br/><a NAME="E8:30"/><i><font color="Green" title="E11">V3</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c7">g</font> <a href="ami_1.html#K11" title="AMI_1:func.11">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c7">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font>)
 <b>by </b><i><a class="ref" href="scmnorm.html#D6" target="_self" title="SCMNORM:def.6">Def6</a></i>;<br/>
<a NAME="E9:30"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">g</font>
 
<b>from </b><i><a class="ref" href="nat_1.html#S16" title="NAT_1:sch.16">NAT_1:sch 16</a>(<a class="txt" href="#E3:30"><i><font color="Green" title="E7">W2</font></i></a>, <a class="txt" href="#E4:30"><i><font color="Green" title="E8">W3</font></i></a>, <a class="txt" href="#E7:30"><i><font color="Green" title="E10">V2</font></i></a>, <a class="txt" href="#E8:30"><i><font color="Green" title="E11">V3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E10:30"/>
 <a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K10" title="SCMNORM:func.10">Following</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="scmnorm.html#K11" title="SCMNORM:func.11">Comput</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c5">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:30"><i><font color="Green" title="E9">V1</font></i></a>, <a class="txt" href="#E8:30"><i><font color="Green" title="E11">V3</font></i></a>, <a class="txt" href="#E2:30"><i><font color="Green" title="E6">W1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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