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<span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a><br/>  for <font color="Olive" title="b2">a</font> being   <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> <br/>  for <font color="Olive" title="b3">Ig</font> being   <a href="scmfsa7b.html#V1" title="SCMFSA7B:attr.1">good</a>  <a href="compos_1.html#NM5" title="COMPOS_1:NM.5">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  st <font color="Olive" title="b1">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">Ig</font>,<font color="Olive" title="b1">s</font>,<font color="Maroon" title="c3">p</font> &amp;  <a href="scmfsa9a.html#R4" title="SCMFSA9A:pred.4">WithVariantWhile&gt;0</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">Ig</font>,<font color="Olive" title="b1">s</font>,<font color="Maroon" title="c3">p</font> holds <br/> for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being    <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>   st <font color="Olive" title="b4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">j</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scmfsa9a.html#K2" title="SCMFSA9A:func.2">ExitsAtWhile&gt;0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">s</font>) &amp; <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scmfsa9a.html#K2" title="SCMFSA9A:func.2">ExitsAtWhile&gt;0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">s</font>) holds <br/>( <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">j</font> &amp;  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">j</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">s</font> be   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> <br/>  for <font color="Olive" title="b2">Ig</font> being   <a href="scmfsa7b.html#V1" title="SCMFSA7B:attr.1">good</a>  <a href="compos_1.html#NM5" title="COMPOS_1:NM.5">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  st <font color="Maroon" title="c4">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">Ig</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">p</font> &amp;  <a href="scmfsa9a.html#R4" title="SCMFSA9A:pred.4">WithVariantWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">Ig</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">p</font> holds <br/> for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being    <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>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scmfsa9a.html#K2" title="SCMFSA9A:func.2">ExitsAtWhile&gt;0</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>) &amp; <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scmfsa9a.html#K2" title="SCMFSA9A:func.2">ExitsAtWhile&gt;0</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>) holds <br/>( <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">j</font> &amp;  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">j</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Ig</font> being   <a href="scmfsa7b.html#V1" title="SCMFSA7B:attr.1">good</a>  <a href="compos_1.html#NM5" title="COMPOS_1:NM.5">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>  st <font color="Maroon" title="c4">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">p</font> &amp;  <a href="scmfsa9a.html#R4" title="SCMFSA9A:pred.4">WithVariantWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">p</font> holds <br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being    <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>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scmfsa9a.html#K2" title="SCMFSA9A:func.2">ExitsAtWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>) &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scmfsa9a.html#K2" title="SCMFSA9A:func.2">ExitsAtWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>) holds <br/>( <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">j</font> &amp;  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">j</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">Ig</font> be   <a href="scmfsa7b.html#V1" title="SCMFSA7B:attr.1">good</a>  <a href="compos_1.html#NM5" title="COMPOS_1:NM.5">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</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="c4">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">p</font> &amp;  <a href="scmfsa9a.html#R4" title="SCMFSA9A:pred.4">WithVariantWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">p</font> implies  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <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>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scmfsa9a.html#K2" title="SCMFSA9A:func.2">ExitsAtWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>) &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scmfsa9a.html#K2" title="SCMFSA9A:func.2">ExitsAtWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>) holds <br/>( <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">j</font> &amp;  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">j</font></span>)</span> ) )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c7">I</font> = <font color="Maroon" title="c6">Ig</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c4">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:50"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
 <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">p</font>
 <span class="kw">and </span><br/><a NAME="E3:50"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
 <a href="scmfsa9a.html#R4" title="SCMFSA9A:pred.4">WithVariantWhile&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">p</font>
 ; <a class="txt" 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>, <font color="Olive" title="b2">j</font> being    <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>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scmfsa9a.html#K2" title="SCMFSA9A:func.2">ExitsAtWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>) &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scmfsa9a.html#K2" title="SCMFSA9A:func.2">ExitsAtWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>) holds <br/>( <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">j</font> &amp;  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">j</font></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">SW</font> =  <a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c9">PW</font> = <font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>)</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">K</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E5:50"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
 <a href="scmfsa9a.html#K2" title="SCMFSA9A:func.2">ExitsAtWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">K</font>
 <span class="kw">and </span><br/><a NAME="E6:50"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">K</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E7:50"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <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>   st <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c10">K</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font>
 <span class="kw">and </span><br/><a NAME="E8:50"/>
 <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>)</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>)</span>)</span></span>)</span>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="scmfsa9a.html#D6" target="_self" title="SCMFSA9A:def.6">Def6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a>  the <a href="memstr_0.html#U1" title="MEMSTR_0:sel.1">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a><span class="kw"> such that </span><br/><a NAME="E10:50"/><span class="lab"><font color="Green" title="E50">A7</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <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>   holds <br/>( <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="scmfsa9a.html#D5" target="_self" title="SCMFSA9A:def.5">Def5</a></span>;<br/>
<a NAME="E11:50"/><span class="lab"><font color="Green" title="E51">A8</font></span>: 
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <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>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> holds <br/> <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">j</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> implies  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50_1"/><span class="lab"><font color="Green" title="E52">A9</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">and </span><br/><a NAME="E2:50_1"/>
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font>
 <span class="kw">and </span><br/><a NAME="E3:50_1"/><span class="lab"><font color="Green" title="E53">A10</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:50_1_1"/>
( <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">K</font> or <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">K</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:50_1_1_1"/><span class="lab"><font color="Green" title="E54">A11</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span>)</span></span><br/><div class="add"><span class="kw">assume </span><a NAME="E2:50_1_1_1"/>
 <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:50_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:50_1_1_1"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="ref" href="scmfsa6a.html#T7" title="SCMFSA6A:th.7">SCMFSA6A:7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:50_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E1:50_1_1_1"><span class="lab"><font color="Green" title="E54">A11</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:50_1_1_2"/><span class="lab"><font color="Green" title="E54">A12</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span>)</span></span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> implies  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span></span>)</span> );<br/><a NAME="E2:50_1_1_2"/><span class="lab"><font color="Green" title="E55">A13</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">k</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50_1_1_2_1"/><span class="lab"><font color="Green" title="E56">A14</font></span>: 
( <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> implies  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E2:50_1_1_2_1"/><span class="lab"><font color="Green" title="E57">A15</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E3:50_1_1_2_1"/><span class="lab"><font color="Green" title="E58">A16</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="scmfsa9a.html#T33" target="_self" title="SCMFSA9A:th.33">Th39</a></span>;<br/>
<a NAME="E4:50_1_1_2_1"/><span class="lab"><font color="Green" title="E59">A17</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/>
<a NAME="E5:50_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A18</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_1_1_2_1"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E6:50_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A19</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E49">A6</font></span></a></span>;<br/>
<a NAME="E7:50_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A20</font></span>: 
<font color="Maroon" title="c6">Ig</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span>,<font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="scmfsa9a.html#D4" target="_self" title="SCMFSA9A:def.4">Def4</a></span>;<br/>
<a NAME="E8:50_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A21</font></span>: 
<font color="Maroon" title="c6">Ig</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a>  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_1_2_1"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="scmfsa9a.html#E51"><span class="lab"><font color="Green" title="E41">Lm7</font></span></a></span>;<br/>
<a NAME="E9:50_1_1_2_1"/><span class="lab"><font color="Green" title="E64">A22</font></span>: 
<font color="Maroon" title="c6">Ig</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span>,<font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E6:50_1_1_2_1"><span class="lab"><font color="Green" title="E61">A19</font></span></a>, <a class="ref" href="scmfsa9a.html#D4" target="_self" title="SCMFSA9A:def.4">Def4</a></span>;<br/>
<a NAME="E10:50_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A23</font></span>: 
<font color="Maroon" title="c6">Ig</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a>  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_1_2_1"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E7:50_1_1_2_1"><span class="lab"><font color="Green" title="E62">A20</font></span></a>, <a class="txt" href="scmfsa9a.html#E52"><span class="lab"><font color="Green" title="E42">Lm8</font></span></a></span>;<br/>
<a NAME="E11:50_1_1_2_1"/><span class="lab"><font color="Green" title="E66">A24</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="scmfsa9a.html#T33" target="_self" title="SCMFSA9A:th.33">Th39</a></span>;<br/>
<a NAME="E12:50_1_1_2_1"/><span class="lab"><font color="Green" title="E67">A25</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:50_1_1_2_1_1"/>
 not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:50_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A26</font></span>: 
<font color="Maroon" title="c10">K</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E49">A6</font></span></a></span>;<br/>
<a NAME="E3:50_1_1_2_1_1"/>
<font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:50_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_1_1_2_1"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="txt" href="#E2:50_1_1_2_1_1"><span class="lab"><font color="Green" title="E68">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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/>


</div><span class="kw">end;</span></div>
<a NAME="E13:50_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A27</font></span>: 
<font color="Maroon" title="c6">Ig</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span>,<font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="scmfsa9a.html#D4" target="_self" title="SCMFSA9A:def.4">Def4</a></span>;<br/>
<a NAME="E14:50_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A28</font></span>: 
<font color="Maroon" title="c6">Ig</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a>  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:50_1_1_2_1"><span class="lab"><font color="Green" title="E66">A24</font></span></a>, <a class="txt" href="scmfsa9a.html#E51"><span class="lab"><font color="Green" title="E41">Lm7</font></span></a></span>;<br/>
<a NAME="E15:50_1_1_2_1"/>
<font color="Maroon" title="c6">Ig</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span>,<font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E12:50_1_1_2_1"><span class="lab"><font color="Green" title="E67">A25</font></span></a>, <a class="ref" href="scmfsa9a.html#D4" target="_self" title="SCMFSA9A:def.4">Def4</a></span>;<br/>
<a NAME="E16:50_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A29</font></span>: 
<font color="Maroon" title="c6">Ig</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a>  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:50_1_1_2_1"><span class="lab"><font color="Green" title="E66">A24</font></span></a>, <a class="txt" href="#E13:50_1_1_2_1"><span class="lab"><font color="Green" title="E68">A27</font></span></a>, <a class="txt" href="scmfsa9a.html#E52"><span class="lab"><font color="Green" title="E42">Lm8</font></span></a></span>;<br/>
<span class="kw">thus </span> <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
 <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

<br/>.= 
 <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><a href="scmfsa6b.html#K2" title="SCMFSA6B:func.2">IExec</a> (<font color="Maroon" title="c6">Ig</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:50_1_1_2_1"><span class="lab"><font color="Green" title="E66">A24</font></span></a>, <a class="txt" href="#E12:50_1_1_2_1"><span class="lab"><font color="Green" title="E67">A25</font></span></a>, <a class="txt" href="#E14:50_1_1_2_1"><span class="lab"><font color="Green" title="E69">A28</font></span></a>, <a class="txt" href="#E16:50_1_1_2_1"><span class="lab"><font color="Green" title="E70">A29</font></span></a>, <a class="ref" href="scmfsa9a.html#T32" target="_self" title="SCMFSA9A:th.32">Th38</a></span>
<br/>.= 
 <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><a href="scmfsa6b.html#K2" title="SCMFSA6B:func.2">IExec</a> (<font color="Maroon" title="c6">Ig</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1_2_1"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="txt" href="#E2:50_1_1_2_1"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="txt" href="#E4:50_1_1_2_1"><span class="lab"><font color="Green" title="E59">A17</font></span></a>, <a class="txt" href="#E3:50_1_1_2_1"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E7:50_1_1_2_1"><span class="lab"><font color="Green" title="E62">A20</font></span></a>, <a class="txt" href="#E9:50_1_1_2_1"><span class="lab"><font color="Green" title="E64">A22</font></span></a>, <a class="ref" href="scmfsa8c.html#T20" title="SCMFSA8C:th.20">SCMFSA8C:20</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>
<br/>.= 
 <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1_1_2_1"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E6:50_1_1_2_1"><span class="lab"><font color="Green" title="E61">A19</font></span></a>, <a class="txt" href="#E8:50_1_1_2_1"><span class="lab"><font color="Green" title="E63">A21</font></span></a>, <a class="txt" href="#E10:50_1_1_2_1"><span class="lab"><font color="Green" title="E65">A23</font></span></a>, <a class="ref" href="scmfsa9a.html#T32" target="_self" title="SCMFSA9A:th.32">Th38</a></span>
<br/>.= 
 <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</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/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c14">p</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E4:50_1_1_2"/><span class="lab"><font color="Green" title="E56">A30</font></span>: 
<font color="Maroon" title="c10">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">p</font>
 <span class="kw">and </span><br/><a NAME="E5:50_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_1_2"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></span>;<br/><span class="kw">assume </span><a NAME="E6:50_1_1_2"/>
 <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E7:50_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A31</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/><a NAME="E8:50_1_1_2"/>
 for <font color="Olive" title="b1">k</font> being    <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E7:50_1_1_2"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="txt" href="#E2:50_1_1_2"><span class="lab"><font color="Green" title="E55">A13</font></span></a>);<br/></span><a NAME="E9:50_1_1_2"/><span class="kw">then </span>
 <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1_1_2"><span class="lab"><font color="Green" title="E56">A30</font></span></a></span>;<br/><a NAME="E10:50_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A32</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">p</font></span>)</span></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="scmfsa6a.html#T7" title="SCMFSA6A:th.7">SCMFSA6A:7</a></span>;<br/><a NAME="E11:50_1_1_2"/>
<font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E4:50_1_1_2"><span class="lab"><font color="Green" title="E56">A30</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E12:50_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E10:50_1_1_2"><span class="lab"><font color="Green" title="E58">A32</font></span></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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E12:50"/><span class="lab"><font color="Green" title="E52">A33</font></span>: 
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <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>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> holds <br/><span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">j</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font> implies <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50_2"/><span class="lab"><font color="Green" title="E53">A34</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">and </span><br/><a NAME="E2:50_2"/>
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font>
 <span class="kw">and </span><br/><a NAME="E3:50_2"/><span class="lab"><font color="Green" title="E54">A35</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">K</font>
 ; <a class="txt" 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="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> implies <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span> );<br/>
<a NAME="E4:50_2"/><span class="lab"><font color="Green" title="E55">A36</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E3:50_2"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:50_2"/><span class="lab"><font color="Green" title="E56">A37</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">k</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50_2_1"/><span class="lab"><font color="Green" title="E57">A38</font></span>: 
( <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k</font> &amp; <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font> implies <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E2:50_2_1"/><span class="lab"><font color="Green" title="E58">A39</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E3:50_2_1"/><span class="lab"><font color="Green" title="E59">A40</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j</font>
 ; <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="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span><br/>

<a NAME="E4:50_2_1"/><span class="lab"><font color="Green" title="E60">A41</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_2_1"><span class="lab"><font color="Green" title="E58">A39</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:50_2_1_1"/>
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> or <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_2_1"><span class="lab"><font color="Green" title="E60">A41</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:50_2_1_1_1"/><span class="lab"><font color="Green" title="E61">A42</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font>
 ; <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="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span><br/><div class="add"><a NAME="E2:50_2_1_1_1"/>
 not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="txt" href="#E4:50_2"><span class="lab"><font color="Green" title="E55">A36</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:50_2_1_1_1"/>
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:50"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="txt" href="#E1:50_2_1_1_1"><span class="lab"><font color="Green" title="E61">A42</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:50_2_1_1_2"/><span class="lab"><font color="Green" title="E61">A43</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k</font>
 ; <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="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span><br/><div class="add"><a NAME="E2:50_2_1_1_2"/><span class="lab"><font color="Green" title="E62">A44</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_2_1"><span class="lab"><font color="Green" title="E59">A40</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:50_2_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:50_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">K</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50"><span class="lab"><font color="Green" title="E49">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:50_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_2"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="txt" href="#E2:50_2_1_1_2"><span class="lab"><font color="Green" title="E62">A44</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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/></div><span class="kw">end;</span></div><a NAME="E4:50_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:50"><span class="lab"><font color="Green" title="E50">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:50_2_1_1_2"/>
<font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2_1"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E3:50_2_1"><span class="lab"><font color="Green" title="E59">A40</font></span></a>, <a class="txt" href="#E1:50_2_1_1_2"><span class="lab"><font color="Green" title="E61">A43</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E6:50_2"/><span class="lab"><font color="Green" title="E57">A45</font></span>: 
<span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:50_2"/><span class="lab"><font color="Green" title="E58">A46</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E8:50_2"/>
 for <font color="Olive" title="b1">k</font> being    <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E7:50_2"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="txt" href="#E5:50_2"><span class="lab"><font color="Green" title="E56">A37</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E9:50_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E6:50_2"><span class="lab"><font color="Green" title="E57">A45</font></span></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/>


</div><span class="kw">end;</span></div>
<span class="kw">given </span><font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">j</font> being    <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> <span class="kw"> such that </span><a NAME="E14:50"/><span class="lab"><font color="Green" title="E53">A47</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">and </span><br/><a NAME="E15:50"/><span class="lab"><font color="Green" title="E54">A48</font></span>: 
( <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scmfsa9a.html#K2" title="SCMFSA9A:func.2">ExitsAtWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>) &amp; <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scmfsa9a.html#K2" title="SCMFSA9A:func.2">ExitsAtWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>) &amp; ( <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font> or  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">Ig</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E16:50"/>
( <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font> or <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:50"><span class="lab"><font color="Green" title="E53">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:50"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E12:50"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="txt" href="#E11:50"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="txt" href="#E15:50"><span class="lab"><font color="Green" title="E54">A48</font></span></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/>


</div>
