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<span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</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">  for <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b2">I</font> being   <a href="compos_1.html#NM4" title="COMPOS_1:NM.4">Program</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> holds <br/> ( <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">while&lt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">I</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">i</font>) <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&gt;=0_goto</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">while&lt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">I</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">I</font></span>)</span> <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="scmpds_2.html#K3" title="SCMPDS_2:func.3">goto</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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">  for <font color="Olive" title="b1">I</font> being   <a href="compos_1.html#NM4" title="COMPOS_1:NM.4">Program</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> holds <br/> ( <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">while&lt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b1">I</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>) <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&gt;=0_goto</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">while&lt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b1">I</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">I</font></span>)</span> <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="scmpds_2.html#K3" title="SCMPDS_2:func.3">goto</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">I</font> be   <a href="compos_1.html#NM4" title="COMPOS_1:NM.4">Program</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</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"> ( <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">while&lt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>) <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&gt;=0_goto</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">while&lt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <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="scmpds_2.html#K3" title="SCMPDS_2:func.3">goto</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c6">i1</font> = (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>) <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&gt;=0_goto</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">i2</font> =  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">goto</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">I4</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>) <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&gt;=0_goto</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">';'</a> <font color="Maroon" title="c5">I</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">J5</font> = <font color="Maroon" title="c5">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">goto</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">FLOOP</font> =  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">while&lt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">I</font>);<br/>
<a NAME="E1:12"/>
 <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">while&lt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">I</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>) <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&gt;=0_goto</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">';'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">I</font> <a href="scmpds_4.html#K3" title="SCMPDS_4:func.3">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">goto</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_4.html#T15" onmouseover="rs('scmpds_4/T15')" onmouseout="rh()">SCMPDS_4:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:12"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">while&lt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>) <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&gt;=0_goto</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_6.html#T7" onmouseover="rs('scmpds_6/T7')" onmouseout="rh()">SCMPDS_6:7</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"> <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">while&lt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <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="scmpds_2.html#K3" title="SCMPDS_2:func.3">goto</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/>

<a NAME="E3:12"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>) <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&gt;=0_goto</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">';'</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_6.html#T6" onmouseover="rs('scmpds_6/T6')" onmouseout="rh()">SCMPDS_6:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:12"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">while&lt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <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="scmpds_2.html#K3" title="SCMPDS_2:func.3">goto</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmp_gcd.html#T6" onmouseover="rs('scmp_gcd/T6')" onmouseout="rh()">SCMP_GCD:6</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/>


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